Structural shape optimization using equivalent static loads transformed from dynamic loads

All the loads in the real world act dynamically on structures. Since dynamic loads are extremely difficult to handle in analysis and design, static loads are utilized with dynamic factors. The dyna mic factors are generally determined based on experiences. Therefore, the static loads can cause problems in precise analysis and design. An analytical method based on modal analysis has been proposed for the transformation of dynamic loads into equivalent static load sets. Equivalent static load sets are calculated to generate an identical displacement field in a structure with that from dynamic loads at a certain time. The process is derived and evaluated mathematically. The method is verified through numerical tests. Various characteristics are identified to match the dynamic and the static behaviors. For example, the opposite direction of a dynamic load should be considered due to the vibration response. A dynamic bad is transformed to multiple equivalent static loads according to the number of the critical times. The places of the equivalent static load can be different from those of the dynamic load. An optimization method is defined to use the equivalent static loads. The developed optimization process has the same effect as the dynamic optimization which uses the dynamic loads directly. Standard examples are solved and the results are discussed

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.

The forging process is the shaping of a workpiece using dynamic loads and typically consists of the multi-step process with the preforming process. Defects such as flash and unfilling occur and the distribution of effective strains is not even in the workpiece after the forging process. An optimized preform is necessary for reduction of the defects in the desired final forging. Optimization of the forging process is nonlinear dynamic response optimization because nonlinearities are involved in the analysis. When the conventional method is utilized in optimization of the forging process, the cost is extremely high due to repeated nonlinear analyses for function and sensitivity calculation. In this paper, the equivalent static loads method for non linear static response structural optimization (ESLSO) is employed to determine the preform shape which leads to reduction of the unfilled area and even distribution of the effective strain in the final forging shape. ESLSO is a structural optimization method where nonlinear dynamic loads are transformed to equivalent static loads (ESLs). ESLs are defined as the loads for linear analysis, which generate the same response field as that of nonlinear analysis. 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 solved using these ESLs and the results are discussed.

All the forces in the real-world act dynamically on structures. Since dynamic loads are extremely difficult to handle in analysis and design, static loads are usually utilized with dynamic factors. Static loads are especially exploited well in structural optimization where many analyses are carried out. However, the dynamic factors are not determined logically. Therefore, structural engineers often come up with unreliable solutions. An analytical method based on modal analysis is proposed for the transformation of dynamic loads into Equivalent Static Loads (ESLs). The ESLs are calculated to generate an identical displacement field with that from dynamic loads at a certain time. The process is derived and evaluated mathematically by using the modal analysis. Since the exact solution is extremely expensive, some approximation methods are proposed. Error analyses have been conducted for the approximation methods. Standard examples for structural design are selected and solved by the proposed method. Applications of the method to structural optimization are discussed. Copyright © John Wiley & Sons, Ltd.

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.

Vehicle crash optimization is a representative non-linear dynamic response structural optimization that utilizes highly non-linear vehicle crash analysis in the time domain. In the automobile industries, crash optimization is employed to enhance the crashworthiness characteristics. The equivalent-static-loads method has been developed for such non-linear dynamic response structural optimization. The equivalent static loads are the static loads that generate the same displacement field in linear static analysis as those of non-linear dynamic analysis at a certain time step, and the equivalent static loads are imposed as external loads in linear static structural optimization. In this research, the conventional equivalent-static-loads method is expanded to the crash management system with regard to the frontal-impact test and a full-scale vehicle for a side-impact crash test. Crash analysis frequently considers unsupported systems which do not have boundary conditions and where adjacent structures do not penetrate owing to contact. Since the equivalent-static-loads method uses linear static response structural optimization, boundary conditions are required, and the impenetrability condition cannot be directly considered. To overcome the difficulties, a problem without boundary conditions is solved by using the inertia relief method. Thus, relative displacements with respect to a certain reference point are used in linear static response optimization. The impenetrability condition in non-linear analysis is transformed to the impenetrability constraints in linear static response optimization.

The equivalent static loads method (ESLM) is a structural optimization method that can consider an analysis method other than linear static analysis. This method defines two separate domains: the analysis domain and design domain. Analysis is performed in the analysis domain, equivalent static loads (ESLs) sets are generated, linear static response optimization is carried out in the design domain using the ESLs and the process iterates until the stopping criteria are satisfied. This method is quite popular and some commercial systems have installed the method. Theoretical foundation of ESLM was validated for linear dynamic response optimization by Park and Kang (J Optim Theory Appl 18:191‑200, 2003). They claimed that when the ESLM process terminates, the optimum solution satisfies the Karush-Kuhn-Tucker (KKT) necessary condition. Some critical issues were raised by Stolpe (Struct Multidiscip Optim 50:921‑926, 2014). He showed that the theoretical results in Park and Kang are not valid. In this paper, the validation process of Park and Kang is amended according to the Stolpe’s corrections. It is shown that the original claim for the KKT condition is valid by adding some mathematical aspects.

Aiming at the problems such as low computational efficiency and long process in dynamic topology optimization, combined with bi-directional evolutional structure optimization, the dynamic optimization problem is transformed into the static optimization problem in multiple load cases. Newmark method is used to analyze the structure dynamics load. The topology optimization model based on equivalent static load is established. On the basis of the equivalent static load, multi-objective dynamics evolutionary structure optimization is studied. And the mathematical model of the dynamic stiffness and natural frequency about the multi-objective optimization is established. The mathematical formula of the sensitivity is deduced. Then a complete optimization iterative process is given. The numerical example illustrates that it is feasible to solve the multi-objective dynamic topology optimization with equivalent static loads. Modal tracking technique is introduced to the multi-objective dynamic topology optimization. The numerical example shows that modal tracking technology can track the required modal in the optimization process.

A humanoid is a robot with its overall appearance based on that of the human body. When the humanoid moves or walks, dynamic forces act on the body structure. Although the humanoid keeps the balance by using a precise control, the dynamic forces generate unexpected deformation or vibration and cause difficulties on the control. Generally, the structure of the humanoid is designed by the designer’s experience and intuition. Then the structure can be excessively heavy or fragile. A humanoid design scenario for a systematic design is proposed to reduce the weight of the structure while sufficient strength is kept. Lower parts of the humanoid are selected to apply the proposed design scenario. Multi-body dynamics is employed to calculate the external dynamic forces on the parts and structural optimization is carried out to design the lower parts. Because structural optimization using dynamic forces directly is fairly difficult, linear dynamic response structural optimization using equivalent static loads is utilized. Topology and shape optimizations are adopted for two steps of initial and detailed designs, respectively. Various commercial software systems are used for analysis and optimization. Improved designs are obtained and the design results are discussed.

Nonlinear dynamic response structural optimization using equivalent static loads

Article history: Received October 6, 2014 Accepted 2 February 2015 Available online 2 February 2015 In this research, the influences of modal properties of a micro cubic satellite on equivalent static loads (due to combination of quasi-static and dynamic loads during launch time) have been studied. The study shows that the magnitude of equivalent static loads can be affected by satellite effective modes and effective mass distribution along natural frequencies. Besides, when to distinct launchers with different dynamic environments candidate for launch, analytical results illustrate that sometimes the equivalent static combined load value can be higher for the launcher with smaller quasi-static loads. This phenomenon is due to effective modal mass distribution of the satellite. Consequently, a higher combined load may be occurred during launch for smaller quasi-static conditions. Furthermore, in separation phase of satellite from the launcher, the satellite modal parameters influence the magnitude of applied shock load in both axial (launch) and lateral directions, importantly. Thus the present study yields reliable input values for separation shock test which considers dynamic properties of the satellite in comparison with some standards and launcher manuals, suggest a rough estimation of shock load, neglecting the satellite structure dynamic behavior. © 2015 Growing Science Ltd. All rights reserved.

A numerical method of structural dynamic shape optimization is presented by using reproducing kernel particle method (RKPM), by which the mesh distortion that exists in shape optimal method based on finite element can be eliminated completely and the optimal model for structural dynamic optimization design is built. The discreteness-based design sensitivity analysis in both natural frequency and dynamic response is proposed by using direct differentiation method and discrete derivatives on the basis of structural dynamic analysis, in which the penalty method is employed into imposing the essential boundary conditions, and the derivatives of shape functions with respect to design variables are derived. The algorithm of dynamic sensitivity analysis is testified by numerical example, and the numerical results obtained are in good agreement with those obtained using semi-analytical method and global finite differences method. Finally, by integrating the algorithm mentioned based on RKPM with parameterized descriptive method of boundary shape, two examples for structural dynamic shape optimization are performed.

Satellites are exposed to simultaneous static and dynamic loads during the launch phase due to launcher acceleration and aerodynamics. Therefore, they are designed and tested to withstand all these loads. From design perspective, a common approach is to combine static and dynamic loads into an equivalent static load which is referred to as quasi-static loads [1]. The validation of the structural design of the satellite is performed by tests and/or analysis. This paper discusses the static load test methods with an emphasis on the acceleration tests performed on vibration test systems.

To address the frequent occurrence of rock burst disasters in areas with wide coal pillars during mining in the western mining area of China, the wide coal pillar area of the Tingnan coal mine in Shanxi Province was used as the research background. Theoretical analysis, numerical simulation, and field tests were used to establish the mechanical criterion and the energy criterion for the dynamic instability of wide coal pillars. The process and mechanism of wide coal pillar dynamic instability under dynamic and static load disturbances were revealed, and a wide coal pillar rock burst prevention and control scheme was proposed. The results indicated that when the load above a coal pillar reached the stress failure index and the energy failure index was met, the coal pillar reached the critical conditions for rock burst. With increasing static load, the stress, energy, and range of the plastic zone all showed increasing trends on both sides of the coal pillar. Under a given dynamic load, the stress and plastic zone range of the coal pillar significantly increased compared to those without a dynamic load. Under a given static load, the greater the dynamic load, the more likely the coal pillar was to undergo dynamic instability. The evolution of coal pillar dynamic instability was divided into three stages: energy accumulation, local instability, and dynamic instability. When the critical stress and energy conditions for coal pillar dynamic instability are exceeded, rock burst will occur. To reduce the static and dynamic loads of coal pillars, a rock burst prevention and control scheme of energy release and load reduction was proposed and applied onsite. The monitoring results showed that this control plan effectively reduced the stress of the coal pillar and the dynamic load generated by the fracture of the overlying rock layer, indicating safe mining in this area of wide coal pillars.

Geosynthetic reinforced and pile-supported (GRPS) provide an economic and effective solution for embankments. To investigate the load-bearing mechanism of the GRPS embankment in loess, experimental and numerical studies under static and dynamic loading are carried out. The characteristics of soil arch effect and tensile membrane effect of GPRS embankment under static and dynamic loading are revealed by analyzing pile–soil stress ratio, tension of geogrid, and stress distribution of pile. The test results show that the pile–soil stress ratio under dynamic loading is reduced by 2.3 compared with static loading. In comparison to static load, the soil arching effect is attenuated under dynamic load, and the stronger the static load soil arching effect, the greater the degree of weakening under dynamic load. In addition, under dynamic loading, the tensioned membrane effect is still effective, but its enhancement is not as pronounced as under static loading. Furthermore, by using the finite element software, the numerical model is developed and validated with the experimental results. The parameter analysis of the load-bearing performance of the GRPS embankment is accomplished using the finite element model as well.