Static Load Method Research Articles

Structural dynamic topology optimization based on dynamic reliability using equivalent static loads

An approach for reliability-based topology optimization of interval parameters structures under dynamic loads is proposed. We modify the equivalent static loads method for non linear static response structural optimization (ESLSO) to solve the dynamic reliability optimization problem. In our modified ESLSO, the equivalent static loads (ESLs) are redefined to consider the uncertainties. The new ESLs including all the uncertainties from geometric dimensions, material properties and loading conditions generate the same interval response field as dynamic loads. Based on the definition of the interval non-probabilistic reliability index, we construct the static reliability topology optimization model using ESLs. The method of moving asymptotes (MMA) is employed as the optimization problem solver. The applicability and validity of the proposed model and numerical techniques are demonstrated with three numerical examples.

Shape optimization of the workpiece in the forging process using equivalent static loads

The forging process, which is the shaping of a workpiece using compressive loads, is a representative plastic manufacturing process and typically consists of a multi-step process with a preforming process. The workpiece shape is an important factor because it influences the quality of the final product. After the forging process, defects such as an unfilled area, flash and crack can occur, and the effective strains may not be evenly distributed. Shape optimization of the workpiece is nonlinear dynamic response optimization because nonlinearities are involved in the analysis of the forging process. Many researches are performed to predetermine the workpiece shape using conventional methods. It is well known that the conventional methods are quite costly due to repeated nonlinear analysis for the calculation of function and sensitivity information. In this paper, the equivalent static loads method for non linear static response structural optimization (ESLSO) is employed to determine the workpiece shape which leads to the desired final shape and even distribution of the effective strain. Equivalent static loads (ESLs) are defined as the static loads for linear analysis, which generate the same response field as that of nonlinear analysis. In ESLSO, the dynamic loads for nonlinear analysis are transformed to ESLs. The ESLs, which have the characteristics of nonlinearities and dynamic loads, are utilized as the loading conditions in linear static response optimization. The design is updated from the results of linear static response optimization using ESLs. Nonlinear analysis is carried out with the updated design, and the process proceeds in a cyclic manner until the convergence criteria of the design variables are satisfied. Two kinds of ESLs are proposed and they are the ESLs for the displacements and the ESLs for the effective strains. Examples of the forging process are formulated and solved.

가상모델로부터 산출된 응력 등가정하중을 이용한 금속 성형품 및 단조품의 형상최적설계

본 논문은 응력 등가정하중을 이용하여 금속제조공정에서 원하는 성형품과 단조품의 최종형상을 얻기 위한 형상최적화 방법을 제안한다. 성형품의 최종형상은 블랭크의 형상에 따라 달라지고 단조품의 최종형상은 빌렛의 형상에 따라 달라진다. 따라서 원하는 형상의 제품을 얻기 위해 구조최적화방법 중형상최적화방법을 적용하였다. 금속성형 공정은 비선형 동적해석을 수행하므로 등가정하중법을 이용한다. 등가정하중법 중 가상모델을 이용한 응력 등가정하중은 등가정하중을 산출하는 새로운 방법으로 재료 특성의 가치를 재정의하여 응력 등가정하중을 계산한다. 본 논문에 포함된 예제를 통해 원하는 제품의 최종형상을 얻기 위한 최적의 블랭크 및 빌렛 형상을 도출하여 제안한 방법의 유용성을 검증한다.

Design flow for the crash box in a vehicle to maximize energy absorption

Vehicle collisions frequently happen at a low speed. Insurance companies and the Research Council for Automobile Repairs both require reduction of repair costs and improvement in occupant safety in a low-speed crash. In order to reduce repair costs, an energy absorbing device such as the crash box is usually installed. The crash box is a thin-walled structure attached between the vehicle bumper structure and the side rail. The determination of the crash box geometry is quite important to absorb the impact energy, since the installation space of the crash box is not very large. In this research, a design procedure to determine the cross-sectional dimensions is proposed to enhance the energy absorption capability of the crash box. The proposed process has two steps. In the first step, the cross-sectional dimensions for the conceptual design are determined in two ways. One is a parameter study using discrete design with an orthogonal array. The cross-sectional dimensions of the crash box are selected among the available cross-sections, such as a circle or a polygon. The cross-sectional dimensions are determined by the analysis of the mean from the discrete design with an orthogonal array. The other is topology optimization, which is performed to determine the cross-section of the crash box to maximize the absorbed strain energy based on the Research Council for Automobile Repairs test conditions. The equivalent static loads method for non linear static response structural optimization is employed to solve the formulated topology optimization problems. The cross sections of the crash box are determined from the results of the conceptual design. In the second step, the detailed design processes are performed by using discrete design with an orthogonal array for the models that are selected in the first step. The detailed shapes of the new crash boxes are determined from the detailed design. The optimization problem for the crash box is formulated considering the geometric constraints of fitting into the given space for the crash box. Three new types of crash box are suggested, with detailed shapes from the proposed design procedure.

지지부가 움직이는 모바일하버용 크레인의 동적 응답 최적설계

모바일하버는 해상에서 컨테이너 상하역 작업을 하는 새로운 해상물류 시스템이다. 모바일하버용 크레인과 같이 해상에서 작업하는 대형 구조물은 파도에 의해 발생하는 지지부의 운동으로 관성력의 영향을 크게 받는다. 따라서 구조물의 안전성에 대한 정확한 검증이 요구되며, 생산 비용을 줄이기 위해 경량화가 매우 중요하다. 이런 요구조건을 위해 동적 응답 최적설계를 수행한다. 등가정하중법은 동하중을 등가정하중으로 변환한 후, 정적 응답 최적화기법을 사용하여 문제를 해결하는 동적 응답 최적설계 방법이다. 지지부의 움직임을 고려한 등가정하중법을 제안하고, 제안한 방법으로 모바일하버용 크레인을 최적화한다.

Topology Optimization for Structures With Nonlinear Behavior Using the Equivalent Static Loads Method

Many structures in the real world show nonlinear responses. The nonlinearity may be due to some reasons, such as nonlinear material (material nonlinearity), large deformation of the structures (geometric nonlinearity), or contact between the parts (contact nonlinearity). Conventional optimization algorithms considering the nonlinearities are fairly difficult and expensive because many nonlinear analyses are required. It is quite difficult to perform topology optimization considering nonlinear static behavior because of the many design variables. In the current element density based topology optimization considering nonlinear behavior, low-density finite elements cause serious numerical problems due to excessive mesh distortion. Updating the material of the finite elements based on the density is considerably complicated because of the relationship between the element density and structural material. The equivalent static loads method for nonlinear static response structural optimization (ESLSO) has been proposed for size and shape optimization. The equivalent static loads (ESLs) are defined as the linear static load sets which generate the same displacement field from nonlinear static analysis. In this research, a new algorithm is proposed for topology optimization considering all kinds of nonlinearities by modifying the existing ESLSO. The new ESLSO can overcome the difficulties which may occur in topology optimization with nonlinear static behavior. A nonlinear static response optimization problem is converted to cyclic use of linear static response optimization with ESLs. Therefore, the new ESLSO can generate results of nonlinear static response topology optimization by using well established nonlinear static analysis and linear static response topology optimization methods. Four structural examples are demonstrated using the finite element method. Different kinds of nonlinearities are involved in each example.

Crashworthiness design optimization using equivalent static loads

Crashworthiness is an issue that should be considered when designing a passenger vehicle to ensure the occupants’ safety in a vehicle accident. Many governments and insurance companies around the world suggest conditions relating to passenger safety in designing vehicles, and regulations that include the conditions have been utilized. The suggested regulations reflect the crashworthiness of structures in order to consider passenger safety. Therefore, these conditions should be used as objective functions or constraints when optimizing a vehicle structure. However, it is difficult to apply gradient-based optimization methods to crash optimization problems because of the large non-linearities of the problems which should be considered in the time domain. The non-linearities and oscillation of the responses make it difficult to calculate the sensitivity information. Therefore, a design method regarding the crash optimization problem needs to be developed. A crash problem should consider the crashworthiness of the vehicle. That is, a design problem should be solved regarding the crash energy conveyed from the outside and injuries to the human body. In the present research, a crashworthiness design optimization method using equivalent static loads that considers the strain energy and injuries of the human body is proposed. The equivalent static loads method for non-linear static response structural optimization (ESLSO) is modified to handle responses imposed on the strain energy and the head injury criterion (HIC) responses. The proposed ESLSO is verified through three practical examples. Design optimization of a crash box and a knee bolster are carried out to maximize absorbed impact energy, and size optimization of a frontal structure of a simplified vehicle is performed to reduce head injury. In verifying the proposed method, traditional optimization methods such as the response surface method are used. The excellence and usefulness of the proposed crashworthiness optimization method are proved by successfully applying it to a crash problem and improving the crashworthiness of the vehicle.

A topological optimization approach for structural design of a high‐speed low‐load mechanism using the equivalent static loads method

SUMMARYIn high‐speed low‐load mechanisms, the principal loads are the inertial forces caused by the high accelerations and velocities. Hence, mechanical design should consider lightweight structures to minimize such loads. In this paper, a topological optimization method is presented on the basis of the equivalent static loads method. Finite element (FE) models of the mechanism in different positions are constructed, and the equivalent loads are obtained using flexible multibody dynamics simulation. Kinetic DOFs are used to simulate the motion joints, and a quasi‐static analysis is performed to obtain the structural responses. The element sensitivity is calculated according to the static‐load‐equivalent equilibrium, in such a way that the influence on the inertial force is considered. A dimensionless component sensitivity factor (strain energy caused by unit load divided by kinetic energy from unit velocity) is used, which quantifies the significance of each element. Finally, the topological optimization approach is presented on the basis of the evolutionary structural optimization method, where the objective is to find the maximum ratio of strain energy to kinetic energy. In order to show the efficiency of the presented method, we presented two numerical cases. The results of these analyses show that the presented method is more efficient and can be easily implemented in commercial FE analysis software. Copyright © 2011 John Wiley & Sons, Ltd.

Universal Equivalent Static Wind Load Estimation for Spatial Structures Based on Wind-Induced Envelope Responses

Wind-resistant design for large-span spatial structures is complex and difficult in practice. Existing wind-resistant design methods were reviewed and the adaptation range of each method was discussed. Though the universal Equivalent Static Wind Loading (ESWL) method could produce a universal wind load distribution that simultaneously reproduces largest load effects for all structural members of cantilevered structures, it is found that the universal ESWL method is not applicable for general space structures, such as shell structures. In this paper, a practical approach to estimate the universal ESWL based on Wind-induced Envelope Responses (WER) is presented for this problem from the structural designer's point of view. Then, the efficiency of the presented approach is illuminated with a numerical example of cylinder single layered shell. Finally, the advantages and the disadvantages of existing approaches, as well as the presented one, are compared at many aspects, such as necessary information of wind load, efficiency, convenience in practice, and so on.

Finite Element Analysis of Air Cooler Using Equivalent Static Method and Response Spectrum Method Respectively

Based on the characteristic of cooler frame, the finite element model of air cooler frame is established using ANSYS software. Equivalent static load method and response spectrum method were used for the seismic performance analysis of cooler frame. The equivalent static load is applied at the center of gravity of various masses using MPC184 rigid beam for transferring the load. The deformation and stress of cooler using equivalent static load method are larger than that of cooler using spectrum response method. For this reason, it can be said that the equivalent static load method is more conservative. At last the deformations and stresses of cooler frame using two methods are checked by ASME AG-1 specification. The results showed that the deformation and stress of cooler frame meet the requirements.

A topological optimization method for flexible multi-body dynamic system using epsilon algorithm

In a flexible multi-body dynamic system the typical topological optimization method for structures cannot be directly applied, as the stiffness varies with position. In this paper, the topological optimization of the flexible multi-body dynamic system is converted into structural optimization using the equivalent static load method. First, the actual boundary conditions of the control system and the approximate stiffness curve of the mechanism are obtained from a flexible multi-body dynamical simulation. Second, the finite element models are built using the absolute nodal coordination for different positions according to the stiffness curve. For efficiency, the static reanalysis method is utilized to solve these finite element equilibrium equations. Specifically, the finite element equilibrium equations of key points in the stiffness curve are fully solved as the initial solution, and the following equilibrium equations are solved using a reanalysis method with an error controlled epsilon algorithm. In order to identify the efficiency of the elements, a non-dimensional measurement is introduced. Finally, an improved evolutional structural optimization (ESO) method is used to solve the optimization problem. The presented method is applied to the optimal design of a die bonder. The numerical results show that the presented method is practical and efficient when optimizing the design of the mechanism.

Topological Optimization Approach for Structure Design of High Speed Mechanisms Using Equivalent Static Loads Method

IC packaging device and other high-speed and high precision equipment require the mechanical structures of the high-speed actuator to minimize inertia under certain stiffness constraints.Optimization approach using equivalent static loads method is recently developed effective for optimal design of flexible multi-body dynamics.The optimization approach is modified according to the dynamic behavior characteristics of high-speed mechanism.The quasi-static analysis with kinetic degrees of freedom is used to solve equations of the discrete structure dynamiss with motion joints;The strain energy element sensitivity is changed by divided element kinetic energy,such that the changes of elements influent stiffness as well as inertial force of the structure;A non-dimensional measurement of elements is proposed to judge the important of an element at any situation The proposed method is used for optimal design of a die bonder of microelectronic packaging device,and three methods are compared,the results show that the quasi-static analysis with kinematic degrees of freedom is important for structures with motion joints.When the speed is very high,the inertia has greater impact of the vibration than the stiffness.Numerical results show that the optimal method of optimal structure design of high-speed mechanism to be effective.

Shape optimization of the initial blank in the sheet metal forming process using equivalent static loads

In the sheet metal forming process, forming the final desired shape is difficult to obtain due to wrinkling, tearing, failure of material, etc. Various conditions of the forming process should be controlled for the desired shape. These conditions are the velocity of the punch, the friction factor, the blank holding force, the initial shape of the blank and others. Many researchers have conducted studies to predetermine the initial blank shape. The structural optimization technique is one of them. Non-linear response structural optimization is required because non-linearities are involved in the analysis of the metal forming process. When the conventional method is utilized, the cost is extremely high due to repeated non-linear analysis for function and sensitivity calculation. In this paper, the equivalent static loads (ESLs) method is used to determine the blank shape which leads to the final desired shape and reduced wrinkling. The ESLs method is a structural optimization method where non-linear dynamic loads are transformed into ESLs, and these ESLs are utilized as external loads in linear response optimization. The design is updated in linear response optimization. Non-linear analysis is performed with the updated design and the process proceeds in a cyclic manner. An optimization formulation is defined for the examples, the formulated problems are solved to verify the proposed method and the results are discussed. Non-linear analysis is performed using the commercial software LS-DYNA, NASTRAN is used for calculating the ESLs and linear response optimization, and an interface program for LS-DYNA and NASTRAN is developed. Copyright © 2010 John Wiley & Sons, Ltd.

Technical overview of the equivalent static loads method for non-linear static response structural optimization

Linear static response structural optimization has been developed fairly well by using the finite element method for linear static analysis. However, development is extremely slow for structural optimization where a non linear static analysis technique is required. Optimization methods using equivalent static loads (ESLs) have been proposed to solve various structural optimization disciplines. The disciplines include linear dynamic response optimization, structural optimization for multi-body dynamic systems, structural optimization for flexible multi-body dynamic systems, nonlinear static response optimization and nonlinear dynamic response optimization. The ESL is defined as the static load that generates the same displacement field by an analysis which is not linear static. An analysis that is not linear static is carried out to evaluate the displacement field. ESLs are evaluated from the displacement field, linear static response optimization is performed by using the ESLs, and the design is updated. This process proceeds in a cyclic manner. A variety of problems have been solved by the ESLs methods. In this paper, the methods are completely overviewed. Various case studies are demonstrated and future research of the methods is discussed.

Efficient Topology Optimization of Large Dynamic Finite Element Systems Using Fatigue

The present paper modifies and extends the recently developed equivalent static load method for the optimization of dynamically loaded linear elastic finite element systems with a huge number of degrees of freedom. In the equivalent static load method, dynamic loads have been transformed into equivalent static loads. This leads to an equivalent static response optimization with multiple loading conditions instead of a dynamic optimization problem. In the present paper, the equivalent static load algorithm is modified and extended by introducing a fatigue analysis in the iterative optimization procedure, where damage is used as suitable termination criterion of the iteration, as well as for the determination of a single and meaningful equivalent static load that leads to maximal damage in the structure. During the evolution process the structure is systematically stiffened by using the solid isotropic microstructure with penalization approach until a user-defined damage level is reached. Three standard examples from literature and an industrial application with a large number of degrees of freedom (600,000) demonstrate the computational efficiency of the proposed method.

등가정하중을 사용한 선형 동적반응 위상최적설계 기초연구

All the forces in the real world act dynamically on structures. Design and analysis should be performed based on the dynamic loads for the safety of structures. Dynamic (transient or vibrational) responses have many peaks in the time domain. Topology optimization, which gives an excellent conceptual design, mainly has been performed with static loads. In topology optimization, the number of design variables is quite large and considering the peaks is fairly costly. Topology optimization in the frequency domain has been performed to consider the dynamic effects; however, it is not sufficient to fully include the dynamic characteristics. In this research, linear dynamic response topology optimization is performed in the time domain. First, the necessity of topology optimization to directly consider the dynamic loads is verified by identifying the relationship between the natural frequency of a structure and the excitation frequency. When the natural frequency of a structure is low, the dynamic characteristics (inertia effect) should be considered. The equivalent static loads (ESLs) method is proposed for linear dynamic response topology optimization. ESLs are made to generate the same response field as that from dynamic loads at each time step of dynamic response analysis. The method was originally developed for size and shape optimizations. The original method is expanded to topology optimization under dynamic loads. At each time step of dynamic analysis, ESLs are calculated and ESLs are used as the external loads in static response topology optimization. The results of topology optimization are used to update the design variables (density of finite elements) and the updated design variables are used in dynamic analysis in a cyclic manner until the convergence criteria are satisfied. The updating rules and convergence criteria in the ESLs method are newly proposed for linear dynamic response topology optimization. The proposed updating rules are the artificial material method and the element elimination method. The artificial material method updates the material property for dynamic analysis at the next cycle using the results of topology optimization. The element elimination method is proposed to remove the element which has low density when static topology optimization is finished. These proposed methods are applied to some examples. The results are discussed in comparison with conventional linear static response topology optimization.

Nonlinear dynamic response structural optimization using equivalent static loads

It is well known that nonlinear dynamic response optimization using a conventional optimization algorithm is fairly difficult and expensive for the gradient or non-gradient based optimization methods because many nonlinear dynamic analyses are required. Therefore, it is quite difficult to find practical large scale examples with many design variables and constraints for nonlinear dynamic response structural optimization. The equivalent static loads (ESLs) method is newly proposed and applied to nonlinear dynamic response optimization. The equivalent static loads are defined as the linear static load sets which generate the same response field in linear static analysis as that from nonlinear dynamic analysis. The ESLs are made from the results of nonlinear dynamic analysis and used as external forces in linear static response optimization. Then the same response from nonlinear dynamic analysis can be considered throughout linear static response optimization. The updated design from linear response optimization is used again in nonlinear dynamic analysis and the process proceeds in a cyclic manner until the convergence criteria are satisfied. Several examples are solved to validate the method. The results are compared to those of the conventional method with sensitivity analysis using the finite difference method.

Nonlinear Dynamic Response Structural Optimization of a Joined-Wing Using Equivalent Static Loads

joined-wing configuration that was published by Wolkovich in 1986 has been studied by many researchers (Wolkovich, J., The Joined-Wing: An Overview, Journal of Aircraft, Vol. 23, No. 3, 1986, pp. 161―178. doi: 10.2514/3.45285). Thejoined-wing airplane is defined as an airplane that incorporates tandem wings arranged to form diamond shapes from both the top and front views. joined wing can lead to increased aerodynamic performances as well as a reduction in the structural weight. However, the joined wing has high geometric nonlinearity under the gust load. gust load acts as a dynamic load. Therefore, nonlinear dynamic (transient) behavior of the joined wing should be considered in structural optimization. In previous research, linear dynamic response optimization and nonlinear static response optimization were performed. It is well known that conventional nonlinear dynamic response optimization is extremely expensive. Therefore, in this research, nonlinear dynamic response optimization of a joined wing is carried out by using equivalent static loads. concept of equivalent static loads is expanded and newly proposed for nonlinear dynamic response optimization. Equivalent static loads are the load sets that generate the same response field in linear static analysis as that in nonlinear dynamic analysis. Therefore, nonlinear dynamic response optimization can be conducted by repeated use of linear response optimization. For the verification of efficiency of the proposed method, a simple nonlinear dynamic response optimization problem is introduced. problem is solved by using both the equivalent static loads method and the conventional method with sensitivity analysis using the finite difference method. procedure for nonlinear dynamic response optimization of a joined wing using equivalent static loads is explained, and the optimum results are discussed.

Une relation simplifiée pour le calcul de la période fondamentale de certains types de bâtiments irréguliers

ABSTRACT Building codes, as Algerian code RPA 99, don't allow the use of the equivalent static load method to design buildings presenting a setback in plan and elevation when submitted to seismic loads. One of the reasons is that this method is based on the fundamental period that is calculated using simplified expressions valid only for regular buildings. The objective of this study is to present a dynamic model for estimate the fundamental period of buildings with one level in elevation presenting a setback. The model is based on a two degrees of freedom system accounting for the upper and the lower part of structure. A simple relation to calculate the fundamental period of such buildings is proposed. This relation can be introduced to different design codes.

Statistical analysis of slow crack growth experiments

A common approach for the determination of slow crack growth (SCG) parameters are the static and dynamic loading method. Since materials with small Weibull module show a large variability in strength, a correct statistical analysis of the data is indispensable. In this work we propose the use of the Maximum Likelihood Method and a Baysian Analysis, which, in contrast to the standard procedures, take into account that failure strengths are Weibull distributed. The analysis provides estimates for the SCG parameters, the Weibull module, and the corresponding confidence intervals and overcomes the necessity of manual differentiation between inert and fatigue strength data. We compare the methods to a Least Squares approach, which can be considered the standard procedure. The results for dynamic loading data from the glass sealing of MEMS devices show that the assumptions inherent to the standard approach lead to significantly different estimates.