Unit Load Method Research Articles

A simple compliance modeling method for flexure hinges

Various types of flexure hinges have been introduced and implemented in a variety of fields due to their superior performances. The Castigliano’s second theorem, the Euler-Bernoulli beam theory based direct integration method and the unit-load method have been employed to analytically describe the elastic behavior of flexure hinges. However, all these methods require prior-knowledge of the beam theory and need to execute laborious integration operations for each term of the compliance matrix, thus highly decreasing the modeling efficiency and blocking practical applications of the modeling methods. In this paper, a novel finite beam based matrix modeling (FBMM) method is proposed to numerically obtain compliance matrices of flexure hinges with various shapes. The main concept of the method is to treat flexure hinges as serial connections of finite micro-beams, and the shearing and torsion effects of the hinges are especially considered to enhance the modeling accuracy. By means of matrix calculations, complete compliance matrices of flexure hinges can be derived effectively in one calculation process. A large number of numerical calculations are conducted for various types of flexure hinges with different shapes, and the results are compared with the ones obtained by conventional modeling methods. It demonstrates that the proposed modeling method is not only efficient but also accurate, and it is a more universal and more robust tool for describing elastic behavior of flexure hinges.

A new multiconstraint method for determining the optimal cable stresses in cable-stayed bridges.

Cable-stayed bridges are one of the most popular types of long-span bridges. The structural behaviour of cable-stayed bridges is sensitive to the load distribution between the girder, pylons, and cables. The determination of pretensioning cable stresses is critical in the cable-stayed bridge design procedure. By finding the optimum stresses in cables, the load and moment distribution of the bridge can be improved. In recent years, different research works have studied iterative and modern methods to find optimum stresses of cables. However, most of the proposed methods have limitations in optimising the structural performance of cable-stayed bridges. This paper presents a multiconstraint optimisation method to specify the optimum cable forces in cable-stayed bridges. The proposed optimisation method produces less bending moments and stresses in the bridge members and requires shorter simulation time than other proposed methods. The results of comparative study show that the proposed method is more successful in restricting the deck and pylon displacements and providing uniform deck moment distribution than unit load method (ULM). The final design of cable-stayed bridges can be optimised considerably through proposed multiconstraint optimisation method.

Spatial force-based non-prismatic beam element for static and dynamic analyses of circular flexure hinges in compliant mechanisms

Abstract This paper presents a spatial force-based non-prismatic beam element modeling approach for circular flexure hinges, which is capable of accurate and efficient modeling of both static and dynamic characteristics of flexure-based compliant mechanisms. The spatial force-based non-prismatic beam element is an improvement and extension of the existing beam elements by considering shear and torsional effects of circular flexure hinge. The new consistent mass matrix formulation was derived using the Unit Load method. A parallel-guided compliant mechanism was taken as an example to validate the efficiency and accuracy of the proposed approach through static and modal analyses with fewer non-prismatic beam elements, and the results are well agreed with ANSYS simulation results with massive 3D solid or 2D shell elements. The developed approach is also available to model arbitrarily shaped flexure hinges in compliant mechanisms.

Bearing Capacity Analysis for Statically Indeterminate Attachment System of Tower Crane Considering Single Limb Instability

The tower cranes need constantly attaching to rise in the construction of high-rise buildings. The tower body which sets attached frames combined with the four-rod-type unilateral statically indeterminate attachment system is a common form. With the attachment height and distance increasing, the attachment rods become more delicate and flexible. It is necessary to check the overall and local stability of the attachment structure. When a single limb instability happens, the reduced-order variation structure brings about the redistribution of internal forces. The anti-buckling capability depends on the structure of variation. To the condition that the single limb instability occurrs on the component which has the weakest stiffness, an exact internal force expression of the structure under composite loads has been deduced in terms of the moment equilibrium method. At the same time, the decoupling support stiffnesses of the attachment device in each direction have been obtained by the unit load method. Based on the refined calculation model, the internal forces are further analyzed under the condition that the instability rod bears the fixed Euler critical force, and the structural strength and stability capacity has been judged. The calculation result proves that the whole structure has great bearing potential after a local buckling.

Geometrical Method for Computing the Deflection of Statically Indeterminate Structures

Determining elastic displacements of statically indeterminate structures by the unit-load method derived from the theorem of virtual work are explained in most structural mechanics textbooks. Few studies have paid attention to the geometrical method. This study introduces the curvature-area method, which is based on the deflected curve caused by loads, temperature or support settlement, to solve the displacement of statically indeterminate structures. Computing the deflection of a point of an indeterminate frame was carried out as an example using the curvaturearea method. At the same time, the results were verified by the unit-load method. It is reasonable agreement between them, which reveals the curvature–area method is an effective method in computing deflections of indeterminate structures.

Surrogate Function of Post-Tensioning Cable Forces for Cable-Stayed Bridges

Inclined stay cables in cable-stayed bridges are post-tensioned in order to counteract the effect of the deck dead load. The evaluation of the post-tensioning forces is tackled in previous literature using either of the iterative unit load method, the force equilibrium method, or by using optimization search methods. All of the above mentioned methods are computationally intensive which hamper their further use in an optimal design algorithm of the bridge. This paper presents a simple surrogate function that can be used to evaluate post-tensioning cable forces in semi-cable stayed bridge under the action of the dead load. The post-tensioning forces for several configurations of the semi-fan cable-stayed bridges are evaluated using genetic algorithms, and are approximated using a simple polynomial function. The developed surrogate functions are used to study the effect of the main span length of the bridge as well as of the pylon height on the distribution and magnitude of the post-tensioning cable forces.

Stress and displacement fields in tensile tests of multidirectional laminates with flexible clamped ends

The tensile test of a multidirectional laminate of strip geometry is analysed assuming that the clamping system imposes elastic restrictions to the rotations at both ends. The problem has five redundant unknowns that are determined by applying Engesser’s second theorem. Thus, the stress field can be obtained including clamping effects. The displacement field is determined by applying Engesser’s first theorem in conjunction with the unit load method. Hygrothermal effects are also included in the analysis. A method for determining the end compliances based on longitudinal strain and out-of-plane displacement measurements is proposed. Experiments are simulated by numerical results corresponding to AS4/3501-6 carbon/epoxy composite. Finally, two examples concerning multidirectional laminates with general coupling are analysed.

Closed-form compliance equations for power-function-shaped flexure hinge based on unit-load method

A new type of flexure hinge, called power-function-shaped flexure hinge, is presented in this paper. Based on the unit-load method, the closed-form compliance equations of the flexure hinge are derived, and the motion accuracy is investigated. These equations are verified through finite element analysis, which shows that the maximum error is less than 15%. Based on the compliance equations, numerical simulations are conducted to discuss the performance of the flexure hinge. The results indicate that, the stiffness of the power-function-shaped flexure hinge decreases as r increases, increases as β increases, and increases as t increases; the displacement along axis y is the dominant parasitic motion; and based on the same l, h and b, the power-function-shaped flexure hinge can rotate more accurately than circular flexure hinge with the same β and V-shaped flexure hinge with the same β and r. The equations obtained can be utilized to design the power-function-shaped flexure hinges, and can achieve higher motion accuracy than the commonly used circular flexure hinge and V-shaped flexure hinge.

Comments on the understanding of the Virtual Work Method

Students studying elementary structural analysis usually experience some confusion in understanding the concept of work. Most of them think that the is because it results from a force system, even though the adjective in the term virtual work describes the and not the forces. In this paper several comments are given on the concept of virtuality in the method. The discussion emphasize that the is rather than the applied unit load or displacement. Excerpts from various text books related to the method for the calculation of deflections using the unit load method, the derivation of Maxwell's Law of Reciprocal Deflections, and Betti's Law are all discussed. Derivations of these methods are carried out using two different approaches in which no forces are introduced and emphasis is made on the fact that the is virtual. The term assumed loading case is proposed to replace “virtual force system”. This paper aims to clarify the concept of virtuality to students learning the method, mainly to prevent the confusion which most students have.

河川水質縦断変化データに基づく平水時浸透流域平均原単位の推定

Unit load method has been widely used in evaluation of pollutant load from watershed to lake or inner bay. However this method has several issues for accurate evaluation of pollutant load. In this study, we present a new approach for evaluation of unit load averaged over infiltration basin under low-flow conditions using longitudinal variations of pollutant load and point source using GIS. For this purpose, we conducted water sampling and discharge measurement in Takasaki River, flowing into Lake Inba-numa. The results indicate that unit load averaged over infiltration basin of Takasaki River in spring and summer was quite lower than that in autumn and winter. The tendency of the seasonal variations of unit load is closely related to land use of plowed field.

Basic Displacement Functions in Analysis of Centrifugally Stiffened Tapered Beams

Introducing the concept of basic displacement functions (BDFs), a new beam element is presented for the analysis of rotating tapered beams. BDFs are obtained using the energy method; i.e., the unit-load method. Following the basic principles of structural mechanics, it is shown that the shape functions and consequently the structural matrices can be derived in terms of BDFs. The method is a combination of flexibility and stiffness methods and is considered as the logical extension of the conventional finite element method. The method is employed for both static and free vibration analyses of different types of tapered beams, and the results compare well with those in the literature. Finally, the effects of the taper ratio, hub radius and rotating speed parameter on the natural frequencies and static deflection are investigated. The results obtained are satisfactory even for only a few elements.

Analysis of Non-Prismatic Timoshenko Beams Using Basic Displacement Functions

Introducing the concept of Basic Displacement Functions (BDFs), an innovative method is presented which yields a mechanical based approach rather than a mathematical one for exact static analysis of arbitrarily tapered Timoshenko beams. Holding pure mechanical interpretations, BDFs are obtained using energy methods such as unit load method. It is shown that exact shape functions and consequently structural matrices could be derived in terms of BDFs. Unlike the most of finite element formulations which are based on prescribed displacement fields, the present finite element takes advantage of a flexibility basis which guarantees the exact interpolation of displacement field along the element and leads to fast convergence of the method in static analysis even with few elements. The method could be easily implemented into any standard displacement-based program. Carrying out numerical examples, the competency of the method in both static and free vibration is verified.

A Method for Design of Oblique Intersection Beams in Roof Structures

To facilitate calculation of a reinforced concrete oblique intersection beams in roof structures, the analysis ignores the interaction between moment and torque of node position, assuming the beam ends as hinge supports. Vertical displacement coefficients were found out by unit-load method, the compatibility equation of displacement was set up by the conditions of equal displacement at nodes, which can finally calculate force diagram of beams, and further design beam section. When designing, it is necessary to note reinforcement calculation should be taken into account the change of beam effective cross-section high caused by longitudinal tension reinforcement layer placed. Compared with the engineering example show that the design method is more simple and accurate, the results can provide a reference for the engineering design and related research.

Transverse shear stiffness of a chevron folded core used in sandwich construction

Using Kelsey et al. (1958) unit load method, upper and lower bounds for the effective transverse shear moduli of a chevron folded core used in sandwich construction are analytically derived and compared to finite element computations. We found that these bounds are generally loose and that in some cases chevron folded cores are 40% stiffer than honeycomb-like cores.

Optimization of tensioning strategy for asymmetric cable-stayed bridge and its effect on construction process

A cable-stayed bridge is a bridge that consists of one or more pylons, with cables supporting the deck. Cable-stayed bridges have come into wide use recently because of their economy, stability, and excellent appearance. It is possible to achieve a uniform moment distribution in the stiffening girders mainly by prestressing the cables, which leads to a more economical design in material and weight than other types of bridges. However, to achieve a more uniform moment distribution is a vague objective, so it cannot be easily defined as the optimization problem. In other words, the minimization of the cost or weight as the objective is not directly related to the optimization of the cable prestressing. Therefore, it has been considered as one of the most important, difficult, and also interesting topics among many researchers and bridge engineers to determine the optimal tensioning strategy and how to apply the prestressing forces of the cables of a cable-stayed bridge. A number of approaches to determine the optimal cable tensions have been proposed in the literature (Negrao and Simoes, Comput Struct, 64:741–758, 1997; Wang et al., Comput Struct, 46:1095–1106, 1993; Agrawal, J Bridge Eng, 2:61–67, 1997; Janjic et al., J Bridge Eng, 8:131–137, 2003). Among these approaches, the unit load method (Janjic et al., J Bridge Eng, 8:131–137, 2003) is considered in this paper because it can take into account the actual construction process while other approaches are based on the configuration of the final structure only. In this paper, “two-step approach” based on the unit load method is newly proposed to find the optimal tensioning strategy especially for the atypical asymmetric bridge under construction, which has continuous deck supported by one pylon and stay cables. Some numerical results will be given to show the validity of the new approach suggested in this paper.

Development of the HSPF-Paddy model to estimate watershed pollutant loads in paddy farming regions

The Hydrological Simulation Program-FORTRAN (HSPF) was modified to simulate watershed pollutant loads in paddy farming regions. Dike height and the Dirac delta function were added to a modified HSPF code (HSPF-Paddy) to incorporate the behavior of ponded water and nutrients in rice paddy fields. HSPF-Paddy was validated by examining its application to a rice paddy plot (3000 m 2) and a watershed (1803 km 2) that contained a significant proportion of rice paddy fields (30%). The model efficiency at the plot scale was very high showing 0.52, 0.93, and 0.93 for ponded water depth, and accumulation of total nitrogen, and total phosphorus, respectively, and simulated values and observed data were in close agreement. Model efficiency was somewhat lower at the watershed scale showing 0.82, 0.78, 0.44, 0.61, and 0.63 for stream flow, temperature, 5-day biochemical oxygen demand, total nitrogen, and total phosphorus, respectively. However, model performance was generally within expectations, and generally model efficiency greater than 0.5 is deemed acceptable considering the complexity of the watershed in terms of size and land use. This comprehensive, dynamic model is applicable to a wide range of scales, from plot to watershed, and can be used to simulate various paddy field and watershed environmental conditions, including loads from point and nonpoint source pollution. It may be a practical alternative to the unit-load method for estimates of watershed pollution loads, especially in paddy farming regions, where nonpoint source pollution loads may be poorly estimated because of highly site-specific conditions. The application of HSPF-Paddy is recommended for watershed management and best management practices (BMPs) evaluation in paddy farming regions.

DRIFT CONTROL OF HIGH-RISE BUILDINGS WITH UNIT LOAD METHOD

The use of displacement participation factors obtained by the unit load method provides an effective drift control tool for high-rise buildings. Structural sensitivity coefficients and displacement participation factors for members in a structure with respect to the lateral displacement to be controlled are computed and used for identifying active members and their corresponding displacement components. The drift control method using the displacement participation factors with a variable linking strategy is formulated into an optimization problem to determine the amount of material to be modified. Using the drift control method, a structural design model for a high-rise building is proposed and applied to one verifying example and two moment resisting frames. Time consuming trial-and-error processes related to the structural design of a high-rise building is avoided by the proposed structural design model. As demonstrated in the examples, the maximum lateral displacements of the examples are reduced by 57·05, 40·0 and 16·36% without changing the total weights of the structures. © 1997 John Wiley & Sons, Ltd.

An Efficient Method to Simulate One-and Two-Dimensional Delamination Growth in Composite Laminates

This paper applies the unit load method to simulate one-and twodimensional delamination growth in composite laminates. A virtual crack closure formulation is incorporated in a finite element program to evaluate the mode-I, mode-II and mode-III strain-energy release rates. Based on the responses of the unit load on the potential cracks, the nodal forces and displacements for the evaluation of strain-energy release rate are solved by a series of linear equations without reanalyzing the finite element model during crack growth. The results show that the use of the unit load method significantly reduces the computational time to simulate the delamination growth in composite laminates.

水平変形に注目した鉄骨骨組の部材断面変更法

Steel structure improvement method focusing on lateral drift using unit load method is presented. The concept of displacement participation factor which should be distinguished from sensitivity analysis is obtained from unit load method. Using the factor, the presented algorithm minimize the frame weight being satisfied with the deflection constrain and stress constrain simultaneously. With the method, this paper shows some example for high rise building, and the analysis result mention that the algorithm can be very useful for structural scheme.

Derivation of shape functions for an exact 4‐D.O.F. Timoshenko beam element

AbstractTimoshenko beam elements have been the subject of numerous publications. The difficulty was that of arriving at a superconvergent element with four degrees of freedom, as is the case for the Bernuli‐Euler classical beam element. Two different approaches are presented here for the derivation of the shape functions. The first is based on the flexibility matrix, where utilizing the unit load method, including the term that accounts for the shear deformations in the virtual work expression, the stiffness matrix is derived. Then, a second method is presented to derive the exact shape functions, directly from the differential equations of the Timoshenko beam theory. The resulting shape functions are the same in both methods.