Unit Load Method Research Articles

Research on a Calculation Method for the Horizontal Displacement of the Retaining Structure of Deep Foundation Pits

The precise calculation and effective control of the horizontal displacement of deep foundation pit retaining structures are critical for foundation pit support design and construction. Based on stress–strain linear elastomer theory and considering the deformation coordination between an enclosure wall and its internal support member, a formula for the redundant restraint force acting on the retaining wall was derived through the unit load method and the principle of elastic superposition. Moreover, a method for calculating the horizontal displacement of the retaining structure of a deep foundation pit was formulated, which is convenient for engineering applications. The method can also be used to calculate the horizontal displacement of cantilevered and anchored retaining structures when the loading conditions of the deep foundation pit and the relevant parameters of the enclosure structure are known. A case study was conducted on a standard section with an excavation width H of 19.3 m and an excavation depth h of 17.8 m. The structural parameters of the enclosure wall, along with the elastic support stiffness coefficient and soil layer parameters of the pit, were inputted into a MATLAB calculation code. Then, four internal support constraint forces Fi and the calculated values for the horizontal displacement of the enclosure wall were obtained after running the code. The calculated curve closely matched the curve of values measured in the field. The horizontal displacements of the top of the wall of several cement–soil gravity enclosure structures mentioned in the literature were also calculated. The results of these calculations were then compared with the measured data and corresponding data from the literature. The examples provided clear evidence demonstrating that the proposed method is highly reliable for calculating the horizontal displacement of deep foundation pit enclosure structures.

Anisotropic thermal expansion based on a novel metamaterial

Metamaterials exhibiting exceptional thermoelastic properties are emerging as promising candidates for scientific and engineering applications. Previous studies on mechanical metamaterials with negative thermal expansion (NTE) have primarily focused on enhancing isotropic NTE, often resulting in a high coefficient of thermal expansion (CTE) but low stiffness that characterizes bending-dominated NTE metamaterials. In this study, we proposed a novel mechanical metamaterial with significant anisotropic thermal expansion behavior and substantial stiffness based on the cross structure, re-entrant structure, and triangular structure with NTE. The programmable CTE and effective elastic modulus were analytically derived using the unit load method and stiffness matrix method, respectively, which were validated by numerical results obtained from finite element analysis (FEA). Drawing inspiration from the body-centered cubic and face-centered cubic strategies, two types of extended three-dimensional metamaterials can be obtained. The results demonstrate that a larger length ratio λ and angle θ3 can enhance the NTE effect, while reducing the Young's modulus. The maximum magnitudes of NTE and positive thermal expansion (PTE) are achieved at approximately θ2 = 90°, resulting in a decreased stiffness and slight negative Poisson's ratio (NPR). By adjusting the angles θ2 of re-entrant structures, six distinct anisotropic thermal expansion behaviors can be obtained in this planar metamaterial. Furthermore, a wider range of isotropic CTE as well as an extended range of anisotropic CTE can be achieved in the present metamaterial compared to its original metamaterials. The proposed metamaterial can be combined with traditional triangle metamaterials to enhance thermoelastic properties. The analysis also provides valuable insights into designing metamaterials with large CTE, optimal stiffness, and lightweight properties simultaneously for engineering applications.

Distributed bending stiffness estimation of bridges using adaptive inverse unit load method

Bending stiffness has important applications in bridge damage detection, model updating, and load bearing capacity assessment. However, accurately predicting bending stiffness is a challenging task. In this paper, a two-phase methodology named the adaptive inverse unit load method (AIULM) is developed for distributed reconstruction of bending stiffness from onboard deflection data. In the first phase, the bending stiffness is roughly estimated by uniform discretization. A global stiffness smoothing method is applied to eliminate nonphysical discontinuities of stiffness at interelement boundaries. Then, the smoothed results are applied in the second phase to define a node optimization function to adaptively adjust the element node distribution. An accurate bending stiffness distribution is provided by performing AIULM analysis again based on optimized discretization. Since only real bridge deflection influence lines (DILs) are utilized in the formulation, the AIULM provides a C0-continuous stiffness distribution without any information on internal force. The adaptive optimization of the node distribution allows the practical application of AIULM to different bridges, especially old bridges. Numerical and experimental validation cases, which demonstrate the excellent predictive capability and practical usefulness of the AIULM, have been performed.

Inverse Unit Load Method for Full-Field Reconstruction of Bending Stiffness in Girder Bridges

Inverse Unit Load Method for Full-Field Reconstruction of Bending Stiffness in Girder Bridges

Напряженно-деформированное состояние трехгранной фермы с неразрезными поясами пятигранного составного профиля

Introduction. The author considers designs of triangular trusses that can be used to design roofs of industrial and public buildings, various combined systems, and also as overpass structures of linear facilities. The design of a spatial truss as part of a patent pending development of a triangular truss cover with non-cutting closed-section chords is presented. The purpose of numerical studies is to estimate the stress-strain state (SSS) of a spanning triangular truss subjected to static nodal load and different arrangement of strut elements as well as to create a verification database for further numerical studies of non-faceted interface nodes. Materials and methods. The numerical research methodology takes into account the application of nodal static loads, modelling the placement of the enclosing structure of the pavement using purlins. The use of the unit load method is aimed at determining the response of the load-bearing system as part of a comparative evaluation and comparison with the data obtained from other research tasks. Results. In the course of numerical studies, data were obtained, characterising the deflected mode of the triangular truss model in terms of force distribution in the rods and vertical displacements of the nodes. Conclusions. Analysis of the obtained results shows that the accepted design model of a triangular truss adequately reflects its deflected mode. Practical application of the numerical research technique on the basis of application of unit nodal loads consists in the possibility of using calculation results in the form of a structured data set required in calculations by the limit states method. The results obtained can be used as a basis for verification of data obtained in further numerical studies of non-faceted node constructions within the framework of the lamellar mathematical model. The presented numerical studies are part of the complex of scientific research into the actual performance of spanning triangular trusses.

Lateral displacement mode of column-supported modular steel structures with semi-rigid connections

The lateral displacement of a column-supported modular steel structure with semi-rigid connections (CSMSS-SRC) owing to lateral loading was investigated to determine the corresponding lateral displacement mode and establish a foundation for a direct displacement-based seismic design method. Theoretical equations for the lateral displacement caused by the bending deformation of beams and columns, axial deformation of columns, and shear deformation of the vertical inter-module connections were theoretically derived using slope–deflection equations, unit-load method, and shear deformation definition, respectively. Two simplified single frame finite element models were then established to verify the theoretical equations and study the effects of the rotational and horizontal shear stiffnesses of the vertical inter-module connections on the lateral displacement of the CSMSS-SRC. The results showed that the proposed lateral displacement equations are highly accurate. Furthermore, although the rotational stiffness of the vertical inter-module connection did not exert an appreciable effect on the lateral displacement of the CSMSS-SRC, its horizontal shear stiffness did. These results are expected to provide a reference for the engineering design of CSMSS-SRC.

Mechanical Performance Study of Beam-Column Connection with U-Shaped Steel Damper.

The article proposes the use of a semi-rigid energy-dissipation connection combined with a U-shaped metal damper to avoid brittle failure of rigid steel beam–column connections under seismic loading. The U-shaped metal damper connects the H-section column and the H-section beam to form a new energy-dissipation connection as an energy-dissipation member. Compared with the existing research, this connection has a stable energy-dissipation performance and great ductility. To clarify the mechanism of energy dissipation, mechanical models under two U-shaped damping deformation modes are established. The calculation formulas for the yield load and stiffness are derived for the corresponding deformation mode using the unit load method. Taking the T-shaped beam–column connection and the application of U-shaped steel damper in the beam–column connection as an example, the mechanical model of the connection is established and the calculation formulas for the yield load and stiffness are derived. At the same time, the connection is subjected to a quasi-static test under cyclic loading. The results show that the hysteretic curve of the test is complete and that the skeleton curve is accurate compared to the theory. The error range of the initial stiffness and yield load obtained by the test and the theoretical formula is kept within 20%, indicating that the theoretical formula is reasonable and feasible. In addition, the correctness of the finite element model is verified by establishing a finite element model and comparing it with the test. The mechanical responses of purely rigid connections and rigid semi-rigid composite connections are compared and analyzed using a multi-story and multi-span plane frame as an example. The results show that the model with semi-rigid connections, compared to the model with rigid connections, avoids the gradual loss of bearing capacity caused by the failure of the connection area of the second floor of the main structure and improves the seismic performance of the main structure.

Theoretical development of a stiffness decomposition method for calculating the displacement of complex bending structures

It is very difficult to provide analytical displacement solutions for complex bending structures, such as beams with variable cross-sections, in structural analysis. The common methods used for such analysis—the direct integration method and the conventional graph multiplication method—have disadvantages of inefficiency and large computational costs. Therefore, a new approach called the stiffness decomposition method was proposed to overcome these shortcomings. The fundamental principle of this new approach was derived based on the unit load method. The general calculation equation of displacement was derived and provided for general n-segment complex bending structures, and an operational procedure for this method was constructed to facilitate its application. Then, the method was applied to two case studies involving classic complex bending structures. The results showed the correctness and effectiveness of the proposed method. The stiffness decomposition method was simpler and more efficient than the other two methods: the number of computations required by the stiffness decomposition method accounted for only 47.4% to 84.0% of the number of computations required by the other methods in the two case studies. The clear mathematical and mechanical derivation of the proposed method makes it easy to understand. Furthermore, the simplicity and practicality of this method make it extensively applicable.

Variable substitution integral method and its application for solving displacement curves of framed structures

Structural displacement curves are very important for framed structures in structural engineering. However, the analytical solutions of the curves cannot be provided by the classical material mechanics method and the conventional structural mechanics one directly. So, a general analytical method, the variable substitution integral method, was proposed mainly based on the unit load method. The basic principle and operation procedure of the method was constructed. Then it was applied to some classical cases in structural analysis. The research results show the correctness and effectiveness of the method, which can figure out displacement curves of kinds of framed structures under various types of internal forces directly and conveniently. It can fully compensate for the shortcomings of the two usual methods. The clear mathematics and mechanics background of the method makes it understood easily. The simplicity and practicality of this method make it be applied in structural engineering extensively in great possibility.

Hanger pre-tensioning force optimization of steel tied-arch bridges considering operational loads

Full design analysis of a bridge usually has lots of load cases and leads to the analysis process, which is extremely time-consuming. To reduce the computational cost, the existing hanger pre-tensioning force optimization approaches for cable-supported bridges always consider only a few critical load cases. This study presents an efficient approach for hanger pre-tensioning force optimization of steel tied-arch bridges, which can include all the operational loads and their combinations based on the code requirements. The proposed method develops a load decoupling approach (LDA) to enhance the computational efficiency of structural analysis in optimization and utilize the particle swarm optimization algorithm to search the global optimal design. The LDA divides the structural analysis procedure into two systems: (1) iteration-varying and (2) iteration-invariant. The former evaluates the structural responses induced by different hanger forces using unit load method (ULM) and the latter solely needs a single finite element (FE) analysis before iteration to calculate the structural responses caused by the other loads. The superposition principle combines the results from the two systems to identify the minimum safety margin of the bridge under the ultimate limit state. A practical bridge is examined to compare the proposed method with three traditional hanger force determination approaches. The obtained results illustrate that the proposed method not only owns high efficiency but also leads to a safer design and has a broader application range.

Variable Substitution Integral Method and Its Application for Solving Displacement Curves of Framed Structures

Structural displacement curves are very important for framed structures in structural engineering. However, the analytical solutions of the curves cannot be provided by the classical material mechanics method and the conventional structural mechanics one directly. So, a general analytical method, the variable substitution integral method, was proposed mainly based on the unit load method. The basic principle and operation procedure of the method was constructed. Then it was applied to some classical cases in structural analysis. The research results show the correctness and effectiveness of the method, which can figure out displacement curves of kinds of framed structures under various types of internal forces directly and conveniently. It can fully compensate for the shortcomings of the two usual methods. The clear mathematics and mechanics background of the method makes it understood easily. The simplicity and practicality of this method make it be applied in structural engineering extensively in great possibility.

Mixed beam formulation with cross-section warping for dynamic analysis of thin-walled structures

This paper presents the formulation of a three-dimensional beam finite element (FE) that accounts for cross-section warping and dynamic inertia effects. The model is the extension of an existing mixed formulation, originally developed for the static analysis of thin-walled beams, to the case of dynamic loading conditions.Four independent fields are considered to derive the element governing equations, i.e. material rigid displacements, strains and stresses and an additional displacement field, describing the out-of-plane warping displacement of the beam cross-sections. The latter is independently interpolated in the element volume by including additional degrees of freedom (DOF) to the nodal translations and rotations classically considered in beam formulations.To obtain a consistent form of the element mass matrix, the cross-section displacement shape functions are computed, relating the generalized cross-section displacement fields to the element nodal variables. In mixed FE formulations, these are not assigned a priori, as in displacement-based approaches, but are derived on the basis of material stiffness and element geometry, together with compatibility conditions. Thus, the Unit Load method is applied to deduce the expressions of the shape functions consistent with the force-based approach, assuming the simply-supported beam as reference element configuration.As opposed to the original FE model, the additional warping DOFs are not condensed-out with the definition of the element quantities but are treated as additional global unknowns. This permits a correct description of the inertia effects and ensures continuity of the warping displacement fields between adjacent FEs.Correlation studies are presented to validate the proposed model and investigate the effects of cross-section warping on the dynamic behavior of thin-walled structures. For selected specimens, the studies compare solutions obtained adopting the proposed beam element with those resulting from shell or brick FE models. Modal decompositions and time-history analyses are conducted, assuming both linear elastic and nonlinear constitutive behavior for the latter.

Deformation behavior at SLS of welded I-section steel beams with longitudinally profiled flanges

Welded I-section steel beams with longitudinally profiled (LP) flanges, in which LP steel plates are utilized as the flanges may be applied to achieve structural optimization. Their special geometry raises challenges in deformation design, which leads to requirement of calculation equations. In this paper, theoretical solutions for mid-span deflections considering the influence of variable flange thickness have been deduced with the unit-load method and mathematical integration, and three different static loading conditions are incorporated. Then the reliability is verified by comparing with finite element (FE) models and test results. Straightforward design formulae are proposed by introducing a coefficient on the basis of formulae of conventional beams, which depends on all independent geometrical parameters and is determined through a nonlinear fitting of abundant theoretical results. It also indicated that remarkable amount of steel could be saved with a certain amount increase of additional deformation. The formulae developed herein for SLS deformation design of LP beams may be applied in practical construction design of building and bridge structures, which may also promote application of such advanced steel products.

Numerical analysis and full‐scale experiment on K‐joint deformations in the crank arms of lattice transmission towers

SummaryAfter completion of the construction process including wire stringing of 1,000 kV ultrahigh voltage (UHV) cat‐head electric transmission towers, the outward horizontal K‐joint deformation of one side crank arm may usually exceed 70 mm. Joint slip effects were ignored in the initial structural design of transmission towers, which induced that the calculated K‐joint displacements are much lower and the real deformation cannot meet with the requirement of the acceptant code. First, the shortcomings of the traditional unit load method used to calculate the displacements of transmission towers connected by bolt joints were analyzed. Second, a coefficient for describing the axial force state of bolted joints was proposed. The connection numbers for different types of members in the crank arms of a UHV cat‐head transmission tower were determined. The traditional unit load method was enhanced by importing the force state coefficient and specifying the detailed connection numbers. The K‐joint displacements calculated by the enhanced formula were compared with the experimental results by full‐scale tower experiment. It shows that the enhanced unit load method is suitable for calculating K‐joint deformations in the crank arms of UHV cat‐head transmission towers. Lastly, contribution weight coefficients of different members for K‐joint displacements of the UHV cat‐head transmission tower were calculated by the enhanced unit load method. For bolt joints connected with the tower members with high contribution weight coefficients, the manufacturing accuracy of bolt holes should be improved, and the bolt clearance should be decreased, which can significantly reduce the K‐joint deformation of UHV cat‐head transmission towers.

Topology optimization of nonlinear structures with damping under arbitrary dynamic loading

This study presents an extended unit load method in which the displacement of a chosen degree of freedom (DOF) in a nonlinear structure under arbitrary dynamic loading is expressed as an integration of mutual strain energy density over a continuum domain. This new integral formulation for the displacement of a chosen DOF is developed by using the virtual work principle and can be used for linear or nonlinear structural behaviours. The integral form of the displacement is then used to develop new formulations for structural topology optimization involving arbitrary dynamic loading using the moving iso-surface threshold (MIST) method. Presented are two specific topology optimization problems with two objective functions: (a) to minimize the peak of a chosen displacement; or (b) to minimize the average power spectral density (PSD) of the chosen displacement over a finite time interval. New MIST formulations and algorithms are developed for solving two damping topology optimization problems of a structure under arbitrary dynamic loading, with or without large displacements, and having cellular damping materials with multi-volume fractions. Several numerical examples are presented to demonstrate the validity and efficiency of the presented unit load method and the MIST formulations and algorithms.

Mechanics based analytical approaches to predict nonlinear behaviour of LSCC beams

This paper presents the details of analytical studies carried out towards the prediction of flexural capacity and loaddeflection behaviour of Laced Steel-Concrete Composite (LSCC) beams. Analytical expressions for flexural capacity of the beams are derived in accordance with the basic principles of conventional Reinforced Concrete (RC) beams, but incorporated with relevant modifications to account for the composite nature of the cross-section. The ultimate flexural capacity of the two LSCC beams predicted using the derived expressions is found to be approximately 20% lower than those obtained due to measurement from experiments. Further to these, two simple methods are also proposed on the basis of unit load method and equivalent steel beam method to determine the non-linear load-deflection response of the LSCC beams for monotonic loading. Upon validation of the proposed methods by comparing the predicted responses with those of experiments and finite element analysis, it is found that the methods are useful to find nonlinear response of such composite beams.

Drift Design Method of Steel Moment Frames by using Column-Beam Strength Ratios and Unit-Load Method

This paper presents the resizing method of columns and beams that considers column-to-beam strength ratios to simultaneously control the initial stiffness and ductility of steel moment frames. The proposed method minimizes the top-floor displacement of a structure while satisfying the constraint conditions with respect to the total structural weight and column-to-beam strength ratios. The design variable considered in this method is the sectional area of structural members, and the sequential quadratic programming(SQP) technique is used to obtain optimal results from the problem formulation. The unit load method is applied to determine the displacement participation factor of each member for the top floor lateral displacement; based on this, the sectional area of each member undergoes a resizing process to minimize the top-floor lateral displacement. Resizing members by using the displacement participation factor of each member leads to increasing the initial stiffness of the structure. Additionally, the proposed method enables the ductility control of a structure by adjusting the column-to-beam strength ratio. The applicability of the proposed optimal drift design method is validated by applying it to the steel moment frame example. As a result, it is confirmed that the initial stiffness and ductility could be controlled by the proposed method without the repetitive structural analysis and the increment of structural weights.

Study on the load capacity of a single-section continuum manipulator

This paper presents a simplified method to establish a mathematical model for predicting the loaded posture of a single-section continuum manipulator with arbitrary shape and size of the cross-section. The method constructs a parameter of equivalent bending stiffness for the manipulator and later considers the single-section continuum manipulator to be an equivalent Euler-beam while discussing the symmetry of the cross-section. Thus, the tip position and posture of the manipulator under load are deduced using the large deflection theory and unit load method. The loading model of the single-section continuum manipulator is verified by experiment, and the result indicates that the error between the theoretical model and the experiment is no more than 6%.

Optimization of Pre-Tensioning Cable Forces in Highly Redundant Cable-Stayed Bridges

Cable-stayed bridges have been developing rapidly in the last decade and have become one of the most popular types of long-span bridges. One of the important issues in the design and analysis of cable-stayed bridges is determining the pre-tensioning cable forces that optimize the structural performance of the bridge. Appropriate pre-tensioning cable forces improve the damaging effect of unbalanced loading due to the deck dead load. Because the cable-stayed structure is a highly undetermined system, there is no unique solution for directly calculating the initial cable forces. Numerous studies have been conducted on the specification of cable pre-tensioning forces for cable-stayed bridges. However, most of the proposed methods are limited in their ability to optimize the structural performance. This paper presents an effective multi-constraint optimization strategy for cable-stayed bridges based on the application of an inverse problem through unit load method (ULM). The proposed method results in less stresses in the bridge members, more stability and a shorter simulation time than the existing approaches. The finite element (FE) model of the Tatara Bridge in Japan is considered in this study. The results show that the proposed method successfully restricts the pylon displacement and establishes a uniform deck moment distribution in the simulated cable-stayed bridge; thus, it might be a useful tool for designing other long-span cable-stayed bridges.

Development and Application of an Analytical Model for Horizontally Curved Bridge Decks

This paper presents a reliable closed form analytical solution without any major simplification which can be used with confidence by designers for the analysis of statically indeterminate horizontally curved bridges idealized as curved girders. The unit load method, a method extracted from the Castigliano's second theorem is used to derive the proposed closed form analytical solution which is then used to obtain the influence line diagram of shear force, bending moment and torsion of the curved bridge girder and it is validated with the published test results and also with the finite element solution of a curved girder. Moreover, an extensive parametric study is conducted using the proposed analytical model to investigate the effect of various parameters such as curvature ratio, span length, number of cells and number of loading lanes on bending moment and torsion of the curved bridges subjected to Australian bridge design loads where the effect of all these parameters except the number of cells is found to be significant. Finally, a comparative study of the behaviour of horizontally curved bridges subjected to loads recommended by three different international bridge design codes is undertaken which concludes that the Eurocode LM1 predicts higher bending moment as well as torsion with respect to other codes.