An analysis of the effects of the leg-spacing on spectral response of offshore structures

The authors have applied a computer-based model to perform spectral analysis of jacket platforms. The analysis is carried out in a consistent manner using the finite element method. The proper estimation of the phasing effects of the ocean waves on structural responses is dependent, inter alia, on the accurate calculation of nodal forces. Using the finite element approach the vector of nodal forces can be most accurately computed taking into account the varying distributed loads along the structural members. The model, developed by the first author, calculates cross-receptances of response for all degrees of freedom that enable the phase effects due to the spatial extent of the structure to be properly accounted for. This model includes the phase effects of wave loadings on the platform and it can thus give a more accurate estimation of response spectra which are generally required for design calculations; this is far more accurate than calculating wave loadings on an equivalent cantilever as is often done.     

Deep residual U-net with input of static structural responses for efficient U* load transfer path analysis

U* index theory is widely used to illustrate the load transfer paths inside an engineering structure. However, the conventional U* load transfer path analysis based on the finite element method is computationally demanding especially for large-scale structures. In this research, a convolutional neural network based on the architecture of residual U-Net is introduced to realize high-efficiency U* estimation of plate-type structures with arbitrary dimensions, boundary conditions, and loading conditions for the first time. Besides the geometrical information of the structures, the static structural responses including the feature maps of nodal displacement and stress are involved in the network input. Different input data combinations are experimented to study how they contribute to the model training. It is noticed that the stress and displacement data can significantly lower the output errors in U* prediction, and the geometrical information helps in noise reduction in U* contour graphs. The proposed method is tested with homogeneous plates and functionally graded plates respectively indicating its remarkable performance in load transfer path prediction. Moreover, this method shortens the U* calculation time by over 95% compared to the conventional finite element method. The improved efficiency of load transfer path analysis greatly facilitates the implementation of structural analysis, design, and optimization.     

Nth-order stiffness sensitivities in structural analysis

This paper presents a general expression for theNth-order stiffness sensitivities in linear elastic frames. It is based on modelling the structure as being composed of unimodal elements. It is shown that the sensitivity of the structural response to the variation of the stiffness of an arbitrary component depends only on the corresponding elemental displacements. These are the nodal displacements due to nodal element loads applied to the structure at the end nodes of the considered element. Therefore, on the basis of one structural analysis we obtain the sensitivity of the structure to the variation of a given stiffness, to any order and for all loading conditions. Partial derivatives with respect to several element stiffnesses are obtained from the elemental displacements of the considered elements. The method is equally applicable to more general finite element models. It requires, however, the preliminary decomposition of the finite elements into their unimodal components.     

悬臂梁大变形的向量式有限元分析

为分析悬臂梁的几何非线性行为,用向量式有限元法将结构离散成质点系以及质点间的连接单元.根据牛顿第二定律得到每个质点在内力和外载荷作用下的运动方程以及悬臂梁在每个时刻的变形用该时刻质点系的运动表示.结合刚架元的节点内力和等效质量得出质点位移的迭代计算公式,采用FORTRAN编制计算程序,对悬臂梁分别承受集中载荷和弯矩下的大变形进行算例分析.计算结果与理论解吻合较好,表明该方法能很好地模拟分析悬臂梁的大变形.     

The analysis of nonlinear cable net systems and their supporting structures

The analysis of large cable net systems is feasible only with the aid of high speed digital computers. However, even computerized design of cable nets requires careful formulation and the selection of efficient solution techniques. This paper describes the solution of a general integrated structural system which includes three-dimensional beam members and cable elements that is useful to the design engineer. This solution is used to analyze an example prestressed cable structure, and results are presented to demonstrate the efficiency and accuracy of different solution techniques and to illustrate the effect of variation of important parameters.NET 3 is the general solution program. Input includes: (1) cable sizes, weights, and stress-strain characteristics; (2) initial coordinate points and cable forces due to prestressing; (3) support conditions for the cable net; (4) applied loads; and (5) data for supporting structure. The initial shape and cable forces are computed with the aid of an auxilliary program, SHAPE, given the coordinates of the cable end points, the initial prestressing forces, and an initial interior coordinate. The cable stress-strain characteristics may be linear or nonlinear. The supporting structure may be replaced with equivalent spring stiffnesses or actual three-dimensional elements.The assumptions used in the solution are: (1) cable elements are straight between nodal points and have no flexural stiffnesses; (2) the roofing and decking provide no stiffness; (3) all loads are applied at the nodal points; and (4) all supporting structural elements are elastic. It should be noted that no displacements of the cable nodal points are neglected. Even at boundaries, the elastic stiffness can be considered. Temperature changes can be analyzed to determine their effect upon the structure.Cable net structures usually have many elements and require relatively large amounts of computer time. In addition many loading and geometric conditions must be considered. To be useful as a practical design tool, it was essential that the program be as efficient as possible and require a minimum of storage. The solution utilizes a variation of the Newton-Raphson method. Initially the tangent-modulus stiffness is used. At each iteration the change in the element force is compared with the change in element force for the previous iteration to determine if the convergence rate is within a prescribed value. If it is acceptable, the stiffness for the previous iteration is used; if unacceptable for any element, tangent stiffnesses for all elements are recalculated. This technique attempts to combine the best features of the Newton-Raphson and the modified Newton-Raphson methods by increasing convergence probability and decreasing solution time.To minimize storage requirements, NET 3 has the capability of utilizing tapes in the problem solution. The storage requirements are determined internally before the solution is initiated. If the required storage exceeds that alloted for the system of equations, the solution is effected through the use of tapes.Output for the cables includes nodal point displacements, nodal point forces, and element forces; for beam elements, forces and moments are given. The solution has been verified by comparison with results by other investigators.An elliptical shaped structure, 220 by 240 ft, has been analyzed with the program to investigate the influence of several parameters. The short direction cables were prestressed, and a uniform load of 40 psf was assumed. The variables investigated include: (1) initial prestress force; (2) degree of stiffness of supporting structure; (3) cable sizes; (4) sag-span ratio; and (5) temperature change.     

A co-rotational formulation for nonlinear dynamic analysis of curved euler beam

A co-rotational finite element formulation for the dynamic analysis of a planar curved Euler beam is presented. The Euler-Bernoulli hypothesis and the initial curvature are properly considered for the kinematics of a curved beam. Both the deformational nodal forces and the inertial nodal forces of the beam element are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory in element coordinates which are constructed at the current configuration of the corresponding beam element. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the effectiveness of the proposed element and to investigate the effect of the initial curvature on the dynamic response of the curved beam structures.     

Load mapping for seamless interface between hydrodynamics and structural analyses

This paper addresses the problem of a seamless interface between hydrodynamics and structural analyses. A pressure distribution on a hydro model computed from seakeeping analysis needs to be transferred to a structural model for evaluating structural strength and its integrity. However, due to the differences in the computation and representation methods for both analyses, the load on the hydro model may not be correctly transferred to the structural model, leading to a different load distribution on the structural model and resulting in some unbalanced force and moment components. In this paper, a method is proposed to solve this problem. A pressure distribution on the hydro model is mapped on the structural model through projection, and force and moment imbalances on the structural model are eliminated through optimization of the nodal forces on the structural model. Moreover, a viscous force distribution along the center of each member of the hydro model is transferred to the nodal forces on the structural model based on the minimum distance measure with resolving any force and moment imbalance. Examples are presented to demonstrate the validity of the proposed method.     

Accuracy test of sensitivity analysis in the semi-analytic method with respect to configuration variables

Configuration optimization is a structural optimization method where the geometrical shape of the structures can be changed during the optimization process. Sensitivity informations are required in the general optimization and quite costly. Especially, they are extemely expensive in the structural optimization where the finite element analysis is utilized. Since the nodal coordinates are regarded as design variables in the configuration optimization, the sensitivities according to the nodal coordinates must be calculated. The characteristics of the configuration optimization is that the transformation matrix in the finite element analysis is a function of design variables. Thus the sensitivity calculation in the configuration optimization is even more complicated. For the efficient sensitivity calculations, various methods have been proposed. They are the analytic method (AM), overall finite difference method (OFD), and semi-analytic method (SM). The semi-analytic method consists of the forward and central difference approximation. This study has been conducted to choose an appropriate method by comparison based on the mathematical and numerical aspects. Some standard structural problems are selected for the evaluations.     

Dynamic response analysis of linear structural systems subject to component changes

A method of analysis is developed for determining transient responses of large multiply-connected structural systems subjected to changes of structural components. A dynamic system is divided into two subsystems: the support which remains unaltered and the branch which is liable to change. The response characteristics of an original system are used as a basis for evaluating the new response of the altered system. The responses of the support interface coordinates due to external excitations on it are called base motion. The equations of motion of the system are formulated using subsystem modal properties such that the base motions are the generalized forcing functions. This enables incorporation of the alternative properties in the new analysis without having recourse to the entire system modal properties. The method is applied to a 16-storey building rigid frame model. The methods gives response results comparable with the conventional integrated system analysis. Approximations due to modal truncation are the same as component mode substitution method.     

Topology optimization in crashworthiness design

Topology optimization has developed rapidly, primarily with application on linear elastic structures subjected to static loadcases. In its basic form, an approximated optimization problem is formulated using analytical or semi-analytical methods to perform the sensitivity analysis. When an explicit finite element method is used to solve contact–impact problems, the sensitivities cannot easily be found. Hence, the engineer is forced to use numerical derivatives or other approaches. Since each finite element simulation of an impact problem may take days of computing time, the sensitivity-based methods are not a useful approach. Therefore, two alternative formulations for topology optimization are investigated in this work. The fundamental approach is to remove elements or, alternatively, change the element thicknesses based on the internal energy density distribution in the model. There is no automatic shift between the two methods within the existing algorithm. Within this formulation, it is possible to treat nonlinear effects, e.g., contact–impact and plasticity. Since no sensitivities are used, the updated design might be a step in the wrong direction for some finite elements. The load paths within the model will change if elements are removed or the element thicknesses are altered. Therefore, care should be taken with this procedure so that small steps are used, i.e., the change of the model should not be too large between two successive iterations and, therefore, the design parameters should not be altered too much. It is shown in this paper that the proposed method for topology optimization of a nonlinear problem gives similar result as a standard topology optimization procedures for the linear elastic case. Furthermore, the proposed procedures allow for topology optimization of nonlinear problems. The major restriction of the method is that responses in the optimization formulation must be coupled to the thickness updating procedure, e.g., constraint on a nodal displacement, acceleration level that is allowed.     

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.     

Arch shape optimization using force approximation methods

The objective of this paper is to provide a method of optimizing areas of the members as well as the shape of both two-hinged and fixed arches. The design process includes satisfaction of combined stress constraints under the assumption that the arch ribs can be approximated by a finite number of straight members.In order to reduce the number of detailed finite element analyses, the Force Approximization Method is used. A finite element analysis of the initial structure is performed and the gradients of the member end forces (axial, bending moment) are calculated with respect to the areas and nodal coordinates. The gradients are used to form an approximate structural analysis based on first order Taylor series expansions of the member end forces. Using move limits, a numerical optimizer minimizes the volume of the arch with information from the approximate structural analysis.Numerical examples are presented to demonstrate the efficiency and reliablity of the proposed method for shape optimization. It is shown that the number of finite element analysis is minimal and the procedure provides a highly efficient method of arch shape optimization.     

Sensitivity of structural response in context of linear and non-linear buckling analysis with solid shell finite elements

The paper is concerned with the sensitivity analysis of structural responses in context of linear and non-linear stability phenomena like buckling and snapping. The structural analysis covering these stability phenomena is summarised. Design sensitivity information for a solid shell finite element is derived. The mixed formulation is based on the Hu-Washizu variational functional. Geometrical non-linearities are taken into account with linear elastic material behaviour. Sensitivities are derived analytically for responses of linear and non-linear buckling analysis with discrete finite element matrices. Numerical examples demonstrate the shape optimisation maximising the smallest eigenvalue of the linear buckling analysis and the directly computed critical load scales at bifurcation and limit points of non-linear buckling analysis, respectively. Analytically derived gradients are verified using the finite difference approach.     

A general formulation of structural topology optimization for maximizing structural stiffness

This paper presents a general formulation of structural topology optimization for maximizing structure stiffness with mixed boundary conditions, i.e. with both external forces and prescribed non-zero displacement. In such formulation, the objective function is equal to work done by the given external forces minus work done by the reaction forces on prescribed non-zero displacement. When only one type of boundary condition is specified, it degenerates to the formulation of minimum structural compliance design (with external force) and maximum structural strain energy design (with prescribed non-zero displacement). However, regardless of boundary condition types, the sensitivity of such objective function with respect to artificial element density is always proportional to the negative of average strain energy density. We show that this formulation provides optimum design for both discrete and continuum structures.     

Optimal cross-section and configuration design of elastic-plastic structures subject to dynamic cyclic loading

This paper shows an optimal design problem with continuum variational formulation, applied to nonlinear elasticplastic structures subject to dynamic loading. The total Lagrangian procedure is used to describe the response of the structure. The direct differentiation method is used to obtain the sensitivities of the structural response that are needed to solve the optimization problem. Since unloading and reloading of the structure are allowed, the structural response is path-dependent and an additional step is needed to integrate the constitutive equations. It can be shown, consequently, that design sensitivity analysis is also path-dependent. A finite element method with implicit time integration is used to discretize the state and sensitivity equations.A mathematical programming approach is used for the optimization process. Numerical applications are performed on a 3-D truss structure, where cross-sectional areas and nodal point coordinates are treated as design variables. Optimal designs have been obtained and compared by using two different strategies: a twolevel strategy where the levels are defined according to the type of design variables, cross sectional areas or node coordinates, and optimizing simultaneously with respect to both types of design variables. Comparisons have also been made between optimal designs obtained by considering or not considering the inertial term of the structural equilibrium.     

Non-linear analysis of structures by the finite element method as a supplement to a linear analysis

The discrete elements of finite dimensions which replace the structural continuum in the finite element method can always be chosen sufficiently small that the linear relations between element deformations and element stresses remain valid to the same degree of approximation as is considered acceptable in the linear theory of elasticity. This observation formed the basis for the treatment of geometrical nonlinearities by Argyris and his co-workers in their natural mode technique [1]and [2].Here we give an alternative development of the theory. The element deformations, linearly related to nodal displacements and rotations in a local coordinate system, are expressed as analytic functions of the nodal coordinates in the global system. Then, for structures with an initially linear behaviour, the stability and postbuckling analysis is developed on the basis of the general theory founded by Koiter [3].The theory is illustrated by the example of frame-structures. The location of the nodal points is defined in terms of the displacement vector, while the orientation of an orthogonal triad attached to each nodal point is described by means of modified angular coordinates of Euler. The accuracy of the analysis is demonstrated for a problem solved analytically by Koiter [5]and verified experimentally by Roorda [4].     

Optimal design of laminated composite beam structures

This work deals with design sensitivity analysis and optimal design of composite structures modelled as thin-walled beams. The structures are treated as a torsion-bending resistant beams. The analysis problem is discretized by a finite element technique. A two-node Hermitean beam element is used. The beam sections are made from an assembly of elements that correspond to flat layered laminated composite panels. Optimal design is performed with respect to the lamina orientations and thickness of the laminates. The structural weight is considered as the objective function. Constraints are imposed on stresses, displacements, critical load and natural frequencies. Two failure criteria are used to limit the structural strength: Tsai-Hill and maximum stress. The Tsai-Hill criterion is also adopted to predict the first-ply-failure loads. The design sensitivity analysis is analytically formulated and implemented. An adjoint variable method is used to derive the response sensitivities with respect to the design. A mathematical programming approach is used for the optimization process. Numerical examples are performed on three-dimensional structures.     

用APDL对悬索桥结构进行有限元分析

为实现大跨桥梁体系的结构分析,验证APDL在分析中的高效性,应用ANSYS的APDL建立大跨悬索桥有限元模型．选用合适的求解器,重点分析结构在活载下的响应,同时计算出恒载、风载、温度及基础位移工况下所产生的各控制截面内力及应力分布．计算结果表明,该方法能有效实现悬索桥等复杂结构的内力计算,并满足工程实践要求,方便设计方案更改后的结构重新分析．     

Cable stretching force optimization of concrete cable-stayed bridges including construction stages and time-dependent effects

This paper presents an optimization algorithm to compute the prestressing forces on concrete cable-stayed bridges to achieve the desired final geometry. The structural analysis includes the load history and geometry changes due to the construction sequence and the time-dependent effects due to creep, shrinkage and aging of the concrete. An entropy-based approach was used for structural optimization and discrete direct sensitivity analysis was used to evaluate the structural response to changes in the design variables. Numerical examples are presented and the results exhibit the importance of considering both the construction stages and the time-dependent effects for adequately predict the bridge behaviour and compute the cable prestressing forces.