Load Response Correlation Research Articles

Classification of equivalent static wind loads: Comparisons and applications

This paper reviews the basic ideas, developments, and codification practices of different calculation methods for equivalent static wind load (ESWL) in the past about sixty years. From the landmark work of Davenport's Gust Loading Factor (GLF), Kasperski's Load-Response-Correlation (LRC), and Katsumura, Tamura's Universal-ESWL (U-ESWL), the development thread of ESWL is clarified. The ESWLs are classified based on the number of target load effects (Single-target or Multiple-target) and on actuality (Realistic or Unrealistic). GLF is representative as a “Single-target Unrealistic” ESWL, which has been adopted in many international codes and standards for building design and is extended into many approaches considering different targets (e.g. base bending moment), and different components (along-wind, crosswind, torsional). The LRC method identifies a realistic wind load distribution that causes one target maximum/minimum load effect, is a representative of “Single-target Realistic” ESWL, is well adopted as a background component ESWL, and is further developed to consider the dynamic resonant effect. The U-ESWL simultaneously reproduces maximum/minimum load effects in multiple structural members with only two sets of load distributions, is a “Multiple-target Unrealistic” ESWL, and has also been developed and employed in a wind load code for roof structures.

Determination of Multiple-target Equivalent Static Wind Load of Large-span Roof Structures based on Clustering Analysis Techniques

Spatial structures in modern society exhibit a city’s distinguishing features and show its strength in building technology. Large-span roof structures are mostly seen among spatial structures for various activities. Large-span roof structures are usually sensitive to wind loads due to their lightness in materials and curved geometric appearance. However, spatial structures are generally designed with many structural members, making it challenging to determine adequate load distributions for structural safety analysis. This paper intends to introduce the concept of the multiple-target equivalent static wind loads and to demonstrate how to reduce the heavy computational burden when the structural designer needs to consider multiple loading effects of the target structure. The wind tunnel test of an elliptical-shaped stadium structure with a flat roof is first conducted to show the fundamental aerodynamic characteristics. The methodologies of the background-component wind force based on the load-response-correlation (LRC) method and the resonant-component wind force based on the inertial force method are then introduced for the specification of single-target equivalent static wind loads. Finally, the clustering analysis technique is adopted to explain the concept of the multiple-target equal static wind loads. A decay index is proposed to indicate how the clustering technique improves the specification of equivalent static wind loads.

Dynamic Windage Yaw Angle and Dynamic Wind Load Factor of a Suspension Insulator String

A simplified calculation method is proposed for determining the peak dynamic windage yaw angle φ ^ of electricity transmission line (TL) tower suspension insulator strings (SISs). According to the rigid-body rule, the geometric stiffness matrix in the calculation of the windage yaw angle φ of SISs is dominated by the average wind loads, while the fluctuating wind loads are the dominant factor in the elastic stiffness. With the average wind state of conductors as the initial calculation condition, the load-response-correlation (LRC) method can be used to determine the fluctuating windage yaw angle φ d and the corresponding equivalent static wind loads (ESWLs). Then, the improved rigid straight rod model, which uses the actual length of conductors rather than the projected length, was used to determine the average windage yaw angle φ ¯ . Through the linear superposition of the horizontal increments of φ ¯ and φ ^ d (the peak value of φ d ), the formulae to calculate the φ ^ of SISs were derived. Additionally, the formulae for the dynamic wind load factor, β c , which is a key factor in designing wind loads for φ , were derived according to the principle of ESWLs, rather than being empirically determined by the Chinese code. Thus, the calculation model regarding the loads and response for the φ of SISs was established, and an actual TL was used to verify the established calculation model. Afterward, the influence of the different engineering design parameters on φ and its β c were analyzed. The parameter analyses show that the wind speed, span, and ground roughness influence the magnitudes of φ ^ and β c , however, the height difference between the two suspension points of the conductors, the nominal height, and the sag-to-span ratio may be neglected in the approximate calculation. Our method offers a new solution to TL design when there are large deformations and small strains.

Reduction of Modal Vibration Effect in Load Identification of a Train Bogie Frame

In order to identify the external applied load on train bogie frame, the load response measured by sensor, e.g. strain gauges, is used in engineering practice. However, it is difficult to model the load-response correlation precisely since the bogie frame system characteristic is complicated, especially, it contains many modal vibration effects. The commonly used linear load-response correlation assumption thus deviates from the truth especially at resonant frequencies. A new approach to reduce the modal vibration effect from the load response based on single degree of freedom assumption is proposed in this article. The approach is applied on real measurement of Beijing Subway train bogie frame and the reduction performance is verified.

DESIGN WIND PRESSURE COEFFICIENTS FOR LOW-RISE GABLE-ROOFED STEEL BUILDINGS

The present paper discusses the wind pressure coefficients for the main wind force resisting systems of low-rise gable-roofed steel buildings, based on a wind tunnel experiment and a two-dimensional frame analysis. The wind pressure coefficients should be determined so that they reproduce the maximum load effects. Here, focus is on the bending moments involved in the members as the load effects. The Load Response Correlation (LRC) method is employed for evaluating the equivalent static wind pressure coefficients. Using the time history of wind pressure coefficients, the maximum load effects were computed for all combinations of frame location and wind direction. The results indicate that the most critical condition occurs on the windward frame in a diagonal wind. The largest bending moment was compared with that predicted from the wind pressure coefficients specified in the Japanese building standards, which are based on the area-averaged mean wind pressure coefficients. Finally, more reasonable wind pressure coefficients for designing the main wind force resisting systems are proposed.

Optimization design of large span portal-rigid steel frame with tapered sections under wind-induced drift constraint

Large span portal-rigid frames with tapered sections in low-rise metal building system are often used in industrial and commercial buildings. The drift of such portal-rigid frames induced by the wind load needs to be smaller than a prescribed limit, and at the same time the frames should be as light as possible for cost-effectiveness. To this end, structural optimization of such portal-rigid frames subjected to the wind-induced drift constraint is required. However, little research for optimization of the large span portal-rigid frame with tapered sections has been reported. This paper proposed a procedure for structural optimization of such portal-rigid frames by adopting the Optimality Criteria (OC) method. Based on an aerodynamic database of the measured wind pressure from a wind tunnel test, several forms of the Equivalent Static Wind Load (ESWL) acting on a large-span portal-rigid frame were evaluated by the Gust Loading Factor (GLF), Load Response Correlation (LRC) and Proper Orthogonal Decomposition (POD) method. The method of virtual-work was utilized to construct the wind-induced drift or displacement. An effective iteration algorithm was derived for the minimum weight optimization of the frame while all wind-induced drift constraints were satisfied. The wind resistant structural optimization for the large-span rigid frame with taped section were conducted when it was bore by different forms of ESWL (GLF based, single objective and multiple objective ESWLs), and their optimized results were multi-compared in the form of the optimized total weight and the height of the optimized structural members. The effects of the connection stiffness at semi-rigid joints and the stiffness of support restraints on the optimized results are also investigated. Finally the global optimality property of the proposed OC method adopted for the optimized case in this paper was also discussed and verified. Optimized results show that the OC optimization procedure proposed in this paper is very effective for the optimization design of large-span portal-rigid steel frames with tapered sections under wind-induced drift constraints.

Load–response correlation–based equivalent static wind loads for large cooling towers

The load–response correlation method has been recognized by the wind engineering community as a useful equivalent static wind load calculation method for structural design of quasi-static structures against strong winds. However, it has been found that the load–response correlation method is less effective to non-linear systems and in situations where load processes are non-Gaussian, such as large cooling towers subjected to strong winds. To validate the applicability of the load–response correlation method to large cooling towers, an aero-elastic model has been designed for a 215-m-high cooling tower in this article, which can simultaneously produce wind loads and wind-induced displacements of the structure according to wind tunnel model tests. Using data measured on the aero-elastic model, the exact results of correlation coefficients between wind loads and structural responses are obtained and validated by a non-linear finite element analysis. By comparing the correlation coefficients measured on the scaled model to the results based on the load–response correlation calculation, it is found that the correlations are much stronger for the load–response correlation calculation than those for the exact wind tunnel measurement. The explanation for this observation is that the non-linearity of the real structure and the non-Gaussian feature of the actual wind loads can weaken the correlations between the wind loads and the structural responses.

Design wind force coefficients for the main wind force resisting systems of open- and semi-open-type framed membrane structures with gable roofs

Design wind force coefficients for the main wind force in open- and semi-open-type framed membrane structures were investigated using wind tunnel experiments. The semi-open-type structure has only one gable wall, while the open-type structure has no gable wall. The enclosed-type structure was also tested for comparison. Experiments were conducted with two kinds of turbulent boundary layers corresponding to open-country and urban terrains. First, the distributions of the wind force coefficients, obtained by the difference between the external and internal pressure coefficients, on each type of structure were measured. Based on these results, the effect of the gable-end configuration on the wind force coefficient distribution was determined. Subsequently, various load effects for each type of structure were computed, and the design wind force coefficients for the main wind force resisting systems were proposed using the load response correlation (LRC) method. Two types of column base conditions, hinged-base and clamped-base, were considered. The axial forces induced in the columns for hinged-base frames and the bending moments at the column bases for clamped-base frames were used to investigate the design wind force coefficients. Furthermore, an estimation method of the design wind force coefficients for different turbulent flows is proposed that accounts for the effects of turbulence intensity on the load effects. Finally, a simplified model of the design wind force coefficient is proposed.

Wind resistant size optimization of geometrically nonlinear lattice structures using a modified optimality criterion method

Lattice structures normally have slender members and light weight, high flexibility and small damping ratios. Hence, their structural behavior may be affected by the geometric nonlinearity and they are prone to large displacements and safety issues due to wind loads. Optimal design of such nonlinear and flexible structures is difficult and complicated. This paper presents an automatic optimizing flow for the integrated wind-induced response analysis and wind-resistant optimal design of linear and non-linear lattice structures based on the modified Optimality Criterion (OC) algorithm. The method for structural dynamic analysis and wind-induced response of lattice structure is developed by combining Proper Orthogonal Decomposition (POD) with Load Dependent Ritz (LDR) vector method, together with the Harmonic Excitation Method (HEM) and Load Response Correlation (LRC) method. Meanwhile the quadratic programming method is used to evaluate the Lagrange multipliers in the modified optimality criterion method. Based on the derived design sensitivity analysis results for the nodal displacements, element stresses and nonlinear critical load factor of linear and nonlinear space frame structures in a companion paper, an automatic optimizing flow is then established by adopting the Application Programming Interface (API) of SAP2000 to integrate the wind-induced response, design sensitivity analysis and wind-resistant optimal design of space frame structural system. The proposed automatic optimizing flow is then applied to the optimized process of a lattice structure under dynamic wind loading. It is found that for the lattice structure, the geometric non-linear analysis and the constraint of nonlinear critical load factor need to be considered in their optimization, meanwhile updating on the time-history of wind loads and Equivalent Static Wind Loads (ESWL) is necessary as result of the variations in the design member size during the optimization process. It is also found that the maximum displacement constraint is the most important factor in the optimization of the lattice structure and maximum stress of structural elements are normally found to be inactive constraints during the optimization procedure. In the case that a low limit for the maximum displacement is pre-set, the nonlinear critical load factor becomes the important constraint. Optimized results from this study would be helpful to those involved in wind resistant analysis and optimization design research for such lattice structures.

An efficient method for universal equivalent static wind loads on long-span roof structures

Wind-induced response behavior of long-span roof structures is very complicated, showing significant contributions of multiple vibration modes. The largest load effects in a huge number of members should be considered for the sake of the equivalent static wind loads (ESWLs). Studies on essential matters and necessary conditions of the universal ESWLs are discussed. An efficient method for universal ESWLs on long-span roof structures is proposed. The generalized resuming forces including both the external wind loads and inertial forces are defined. Then, the universal ESWLs are given by a combination of eigenmodes calculated by proper orthogonal decomposition (POD) analysis. Firstly, the least squares method is applied to a matrix of eigenmodes by using the influence function. Then, the universal ESWLs distribution is obtained which reproduces the largest load effects simultaneously. Secondly, by choosing the eigenmodes of generalized resuming forces as the basic loading distribution vectors, this method becomes efficient. Meanwhile, by using the constraint equations, the universal ESWLs becomes reasonable. Finally, reproduced largest load effects by load-response-correlation (LRC) ESWLs and universal ESWLs are compared with the actual largest load effects obtained by the time domain response analysis for a long-span roof structure. The results demonstrate the feasibility and usefulness of the proposed universal ESWLs method.

Coupled wind-induced responses and equivalent static wind loads on long-span roof structures with the consistent load–response–correlation method

Long-span roof structures with multiple vibration modes may undergo coupled motions when exposed to spatiotemporally varying dynamic wind loads. This article presents a framework for the analysis of coupled wind-induced responses and equivalent static wind loads on long-span roof structures. This framework takes into account the inter-modal coupling of modal response components and cross correlation between the background and resonance. The load–response–correlation method is widely used for the background equivalent static wind loads of rigid structures. Also, the concept of load–response–correlation was extended for the correct estimation of equivalent static wind loads by considering the correlation between the load and exact total responses. However, there is no detailed study for the correct estimation of equivalent static wind loads on the component of background, resonance, and their cross response using load–response–correlation method. In this article, a consistent load–response–correlation method for equivalent static wind loads is presented in a unified framework which contains the background, resonance, and modal coupling. First, the accuracy of wind-induced responses is ensured by considering the modal coupling. Meanwhile, the efficiency is improved by decomposing the covariance matrix of the generalized resuming forces. Second, several effects of the wind-induced responses are discussed in detail, such as the effect of the modal coupling, the effect of the modal participations to resonant responses, and the effect of the cross correlation between the background and resonance. Finally, the proposed equivalent static wind loads are represented by the composition of the external wind loads and inertial forces, which are both the most probable load distributions. Accordingly, the equivalent static wind load distribution and the responses by equivalent static wind loads are reasonable.

Assessment of extreme value overestimations with equivalent static wind loads

The wind-resistant design using equivalent static wind loads is convenient for structural engineers. This paper studies the reliability of such an approach in the case of non-Gaussianities in both aerodynamic pressures and responses. These non-Gaussianities are responsible for overestimations of envelope values and may result in uneconomical designs, if not appropriately understood, assessed and addressed. In this study, it is shown that the equivalent static wind loads defined with the Conditional Expected Load method, which extends the physical meaning of the Load-Response Correlation approach in a non-Gaussian framework, improves the issue of overestimations of envelope values. Several envelopes of structural responses are considered: the mean of extremes and the 86% quantiles of extremes, together with two reference periods (10 min and 1 h). Extensive wind tunnel measurements have been collected, which correspond to 371 h full scale. This study is undertaken for quasi-static analysis of structures and is illustrated with a low-rise building.

Reconstruction of the envelope of non-Gaussian structural responses with principal static wind loads

In current practice, structural engineers commonly focus on the wind-resistant design by means of static wind loads. In the case of non-Gaussianities, there is room for improvement to properly derive these static loads. First, this paper extends in a non-Gaussian context the concept of the load-response correlation (LRC) method establishing equivalent static wind loads (ESWLs). This is done by a proper recourse to the new concept of conditional expected static wind load and a proposed bicubic model for the joint and conditional distribution functions. Second, this paper investigates the envelope reconstruction problem targeting the efficient reconstruction of the envelope values of a set of non-Gaussian structural responses by means of principal static wind loads (PSWLs). They have been introduced in a Gaussian context and are obtained by a singular value decomposition of ESWLs. This paper addresses the extension of PSWLs to non-Gaussian structural responses, as well. The developments apply to structures with a linear behavior and subjected to an aerodynamic pressure field exhibiting mildly to strongly non-Gaussian features. In this context, the well-known load-response correlation and conditional sampling methods are used for comparisons. This study is undertaken for quasi-static analysis of structures and is illustrated on a low-rise building.

Equivalent Static Wind Loads on Low Rise Buildings

The many low rise roof structures are sensitive to the effects of fluctuating wind load. In engineering design for the structures, spatiotemporally varying wind loads on the low rise roofs are modeled as equivalent static wind loads. In this paper, the equivalent static load of the large span roofs is formulated in terms of either a weighted combination of modal inertial load components, and the resonant and background load components that was obtained by the POD (Proper Orthogonal Decomposition) and LRC (Load –Response -Correlation) techniques.

Principal static wind loads

In current wind design practice, static wind loads are usually defined to obtain, by simple static analyses, the extreme values of any structural response that would be formally obtained with a strict dynamic buffeting analysis. The minimum and maximum values that may reach any response define the envelope. Equivalent static wind loads (ESWLs) allow to recover extreme responses in the envelope. As a first objective, this paper formalizes a general method to determine ESWL, in a nodal basis, by extending the concept of load–response correlation, which is only valid in the background range. The general method, the displacement–response correlation (DRC) method, covers the background and resonant contributions of the considered response. As a second objective, the paper addresses the problem of building a set of static wind loads that adequately reconstructs the envelopes of responses. The concept of principal static wind loads (PSWL) is introduced to form a reduced basis of representative loads well-suited for envelope reconstruction. Its optimality is demonstrated both analytically and with a detailed illustrative example.

A new methodology for analysis of equivalent static wind loads on super-large cooling towers

For tall buildings and typical long-span spatial structures, the background and resonant components of wind-induced fluctuating response should be taken into account by different calculation theories. The total fluctuating response is obtained through the square root of sum of squares (hereafter referred to as SRSS) combination (hereafter referred to as tri-component method). However, this method cannot consider the modal coupling effects of the background and resonant components, nor the coupling effects of the resonant component for super-large cooling towers. This paper presents a new approach for analyzing wind-induced responses and corresponding equivalent static wind loads (hereafter referred to as ESWLs) by a consistent coupling method (hereafter referred to as CCM) based on structural random vibration theory. Firstly, the refined definition of the cross term between background and resonant component is explained based on a mode-acceleration method, and covariance matrices of coupled elastic restoring force and resonant elastic restoring force are proposed. Secondly, based on covariance matrix theory, CCM is proposed for calculating the background and resonant components and for compensating the cross term between background and resonant components, and the ESWLs of all components are derived by load response correlation theory. Finally, calculation of wind-induced responses and ESWLs for a super-large cooling tower 215m high demonstrates the superiority and effectiveness of the present approach, the characteristics of ESWL distributions of background, resonant and cross term between background and resonant component, and the wind-induced coefficients for super-large cooling towers are extracted.

Grouping response method for equivalent static wind loads based on a modified LRC method

Wind loading is one of the most important loads for controlling the design of large-span roof structures. Equivalent static wind loads, which can generally aim at determining a specific response, are widely used by structural designers. A method for equivalent static wind loads applicable to multi-responses is proposed in this paper. A modified loadresponse- correlation (LRC) method corresponding to a particular peak response is presented, and the similarity algorithm implemented for the group response is described. The main idea of the algorithm is that two responses can be put into one group if the value of one response is close to that of the other response, when the structure is subjected to equivalent static wind loads aiming at the other response. Based on the modified LRC, the grouping response method is put forward to construct equivalent static wind loading. This technique can simultaneously reproduce peak responses for some grouped responses. To verify its computational accuracy, the method is applied to an actual large-span roof structure. Calculation results show that when the similarity of responses in the same group is high, equivalent static wind loads with high accuracy and reasonable magnitude of equivalent static wind distribution can be achieved.

Study of Equivalent Static Wind Loads on a High-Rise Building with 204 Meters High

Equivalent static wind loads (ESWL) are the bridge between engineer in the structure design field and researcher in wind engineering field. In this paper, a practical high-rise building with 204 meters high was tested in wind tunnel, and then ESWL was calculated by load-response correlation (LRC) method based on the results of wind tunnel. Some results are useful for design of building structures.

Distribution of background equivalent static wind load on high-rise buildings

In this paper, the along-wind and cross-wind fluctuating load distributions along the height of high-rise buildings and their correlations are obtained through simultaneous pressure measurements in a wind tunnel. Some typical methods proposed in some relative literatures, i.e., load-response correlation (LRC), and quasi-mean load (QML) and gust load envelope (GLE) methods, are verified in terms of their accuracy in describing the background equivalent static wind load distribution on high-rise buildings. Based on the results, formulae of the distribution of background equivalent static load on high-rise buildings with typical shapes are put forward. It is shown that these formulae are of high accuracy and practical use.

Equivalent static wind loads on low-rise buildings based on full-scale pressure measurements

This paper addresses the modeling of equivalent static wind loads (ESWLs) on low-rise buildings based on full-scale pressure measurements. The wind load effects of different building frames under alongwind excitations are quantified and compared with those based on the wind loads specified in ASCE 7-05 to examine the adequacy of current design wind load. The ESWLs causing different peak wind load effects of building frames are investigated using a number of approaches, which include the gust response factor (GRF), gust loading envelope (GLE) and load–response-correlation (LRC) approaches. This investigation points out the challenges faced in seeking advanced ESWLs on low-rise buildings which are physically meaningful and insensitive to individual response. Finally, a new approach for modeling ESWLs based on the proper orthogonal decomposition of wind loading field is proposed. The new approach provides a set of ESWLs associated with a small number of important loading modes which are independent of individual response and structural systems, and is able to give accurate prediction of all responses. The proposed approach offers a physically meaningful and convenient modeling of wind loads on low-rise buildings with great potential of design applications.