Nodal Equilibrium Equations Research Articles

Prestress state, parametric analysis study and genetic algorithm-based tentative design of a novel pentagonal cable dome structure with tri-strut layout

To improve the structural performance and economy of the large-span roof structure from the structural system level, a novel structural form of pentagonal cable dome with tri-strut layout and large central opening is proposed, which can be applied to the large-span stadium ring canopy structure. Different from Fuller's traditional conception of the tensioning whole, this type of system has three struts intersecting at the same chord node, which reduces the amount of ring and diagonal cables, facilitates tensioning construction, and improves overall stability. For this large-opening cable dome, a general formula for calculating the internal force of prestressed state rods was derived based on the nodal equilibrium equations, and the effects of several parameters on the distribution of pre-tension in the cable dome were analyzed and studied to understand the distribution pattern and characteristics of the cable dome pre-stress. A large-opening stadium canopy with a span of 100m is taken as an example. The parameter sensitivity and correlation analysis of the structural performance reveals that the main influence of the economy of this cable dome is the pre-tension level. In contrast, the geometric parameter produces a much lower effect on the structural economy. The optimal design and trade-off analysis of this cable dome structure based on two genetic algorithms show that appropriately increasing the structural vector height and decreasing the thickness while satisfying the stability improves the structural stiffness and economy. The analysis results show that this new type of cable dome has superior technical and economic indicators, and the research in this paper provides a new form and new ideas for the analysis and design, optimization research, and modeling of the cable dome.

Analytical Study of Structural Conformation and Prestressing State of Drum-Shaped Honeycomb Quad-Strut Cable Dome Structure with Different Calculation Methods

Building upon the analytical study of the structural configuration and prestress state of the drum-shaped quad-strut cable dome, we conducted further investigation into its structural configuration. By employing the nodal equilibrium equations to solve the prestress state analysis of the cable dome, we compared the effects of two different cable laying methods on the prestress state of the cable dome structure. These methods include equal length of the radial horizontal projection of the upper chord ridge cables and equal radial chord length of the upper chord ridge cables. The analysis results show that the radial length of the top chord and its corresponding radial horizontal projection length of the cable dome structure can effectively reflect the trend of the prestress state of the structural configuration. Furthermore, by using a rise-to-span ratio of 0.11 as a threshold, the cable dome configuration is categorized into the flat spheroidal structural configuration and the small hemispheroidal structural configuration. When the structure is analyzed using a small rise-to-span ratio, the difference in prestress calculations between the two structural configurations is found to be less than 10%. Additionally, the structure exhibits a more uniform distribution of prestress, with the prestress gradually increasing from the inner circle to the outer circle. However, when the rise-to-span ratio increases, the difference between the prestress calculation results of the two configurations also increases, emphasizing the need to deploy upper chord ridge cables based on equal radial chord lengths (arc lengths). The research presented in this paper provides a novel insight into the structural topological form and prestress state calculation of cable domes with this configuration.

A strain energy-based homogenization method for 2-D and 3-D cellular materials using the micropolar elasticity theory

A new strain energy-based method for homogenization of 2-D and 3-D cellular materials is developed using an extended version of Hill’s Lemma for the non-Cauchy continuum satisfying the micropolar elasticity theory. Both kinematic and mixed boundary conditions (BCs) that obey the Hill-Mandel condition are identified. The newly proposed method requires that the nodal equilibrium equations be explicitly satisfied at all boundary and interior nodes, unlike the existing approach which does not enforce the equilibrium conditions at interior nodes. For 2-D cellular materials, it is shown that purely kinematic BCs (with prescribed displacement and micro-rotation vectors) and mixed BCs (with prescribed couple stress traction and displacement vectors) can both be used for homogenization, and periodic constraints can be readily accommodated in the former. For 3-D cellular materials, it is demonstrated that mixed BCs (with prescribed couple stress traction and displacement vectors) enable the homogenization satisfying the Hill-Mandel condition. Two sample cases of homogenization of cellular materials are studied by applying the new method – one for a 2-D lattice structure and the other for a 3-D pentamode material. For the 2-D lattice material, the effective classical and micropolar stiffness tensors are obtained using purely kinematic BCs (with and without periodic constraints) and mixed BCs. The closed-form expressions of the effective stiffness tensor components derived here are compared to those given by the existing strain energy-based homogenization method and are seen to be more accurate. For the 3-D pentamode material, mixed BCs are employed in the homogenization, and the effective stiffness tensor components are obtained in closed-form expressions for the first time. The numerical results for the pentamode material given by these analytical formulas are found to agree well with those provided by a finite element model constructed using COMSOL.

Large deformation axial element for implicit constitutive relations

This work documents the formulation of an axial element that could handle implicit constitutive relations. This formulation uses nodal displacements, member forces, and reaction forces as primary unknowns. For some assumed values of these primary unknowns satisfying the nodal equilibrium equations, the computed stretch and stress in the members need not be in agreement with the constitutive relation. Hence, these primary unknowns are found such that the error in the constitutive relation for the members are minimized subjected to the constraint that the nodal equilibrium equations hold. Axial element developed using this concept is used to analyze a few trusses. The results of this formulation are in agreement with the results obtained from ABAQUS®when stress is an explicit function of the stretch ratio. Also, some statically determinate as well as indeterminate trusses are solved for an implicit constitutive relation and the reasonableness of the obtained results established.

Static Damage Identification of 3D and 2D Frames

A new algorithm for static damage detection of three- and two-dimensional frames is presented in this paper. This approach is based on the minimization of difference between the measured and analytical static displacements of frames. The damage detection problem is solved as a nonlinear constrained structural optimization. In this strategy, the global structural stiffness matrix is parameterized. To achieve the goal, a new technique based on the eigen decomposition of the local elemental stiffness matrix is suggested. Structural damage is modeled as a reduction in cross-sectional properties of the elements. It is assumed that the stiffness matrix of the structure is perturbed due to damage. Hence, the damaged structural stiffness matrix is presumed to be the sum of the stiffness matrix of the undamaged structure and the perturbation matrix. Consequently, the sum of these matrices should be inverted in each iteration. Instead of the common ways of inversion, Sherman–Morrison–Woodbury formula is employed. Prior to the outset of the optimization procedure, the number of design variables, the amount of reductions in cross-sectional properties, is decreased by introducing a new suggested technique. This scheme is based on the nodal equilibrium equations. To illustrate the robustness and efficiency of the authors’ algorithm, several numerical problems are solved.

A displacement based optimization method for geometrically nonlinear frame structures

An extension of the displacement based optimization method to frames with geometrically nonlinear response is presented. This method, when applied to small-scale trusses with linear and nonlinear response, appeared to be efficient providing the same solutions as the classical optimization method. The efficiency of the method is due to the elimination of numerous finite element analyses that are required in using the traditional optimization approach. However, as opposed to trusses, frame problems have typically a larger number of degrees of freedom than cross sectional area design variables. This leads to difficulties in the implementation of the method compared to the truss implementation. A scheme that relaxes the nodal equilibrium equations is introduced, and the method is validated using test examples. The optimal designs obtained by using the displacement based optimization and the classical approaches are compared to validate the application to frame structures. The characteristics and limitations of the optimization in the displacement space for sizing problems, based on the current formulation, are discussed.

Conceptual issues in geometrically nonlinear analysis of 3D framed structures

This paper aims to clarify some of the conceptual issues which are related to the geometrically nonlinear analysis of 3D framed structures, and which have been a source of previous confusion. In particular, the paper discusses the symmetry of the tangent stiffness matrix and the nature of the element end moments. It is shown that a symmetric tangent stiffness matrix can always be achieved for a conservative system if the nodal equilibrium equations, including the equations which describe moment equilibrium, are identical to those derived from a variational energy approach. With regard to the element end moments, it is suggested that any definition can be adopted in formulating the geometrically nonlinear element response. Furthermore, it is proposed that any definition for nodal rotations expressing a unique vector transformation may be adopted without compromising modelling accuracy. The argument of this paper is validated with reference to three variants of a large displacement analysis method for 3D frames, where several illustrative examples are utilised.

Exact Static Solution of Grillwork with Periodic Supports

The rectangular grillwork with simply supported boundary condition and periodically distributed internal supports is investigated by using the double U-transformation technique. The equivalent grillwork with cyclic periodicity in two (x and y) directions can be produced by using the image method. The equilibrium equations of nodes of the equivalent grillwork possess cyclic periodicity in the two directions. In the process of analysis for the equivalent grillwork, the double U-transformation is used twice. First, by applying the double U-transformation once, the simultaneous equations for the support reactions are derived. Second, by applying the double U-transformation again, the simultaneous equations can be uncoupled into a set of single-degree-of-freedom equations, thus leading to the exact solution. As an example, a square grillwork with a 6-by-6 mesh and 2-by-2 internal supports is worked out by means of the formulas derived in this study. It can be proven that the explicit closed-form results satisfy the nodal equilibrium equations, the support conditions, and the boundary conditions exactly.

A computer aided analysis of horizontally curved slab-type bridge deck continuous over discrete pier supports

A computer aided method of analysis has been presented for continuous, curved slab-type bridge decks by incorporating the method of finite difference in conjunction with the method of consistent deformation. Using a Levy-type series formulation, equilibrium difference equations are written only along the central radial line of the simply supported deck, by removing the discrete, redundant pier supports. Using the method of consistent deformation, the pier support reactions are determined and hence the entire support structure is solved. The computational procedure based on this one-directional finite difference method is repetitive and obviates the need of a large number of nodal equilibrium equations spread throughout the plate.

Characteristics of Cable Nets

The paper is concerned with the behavior of cable nets. The initial stability of these nets is investigated and the effect of an external load applied to these nets is studied. It is shown that the behavior of cable nets is conditioned by the nature of the nodal equilibrium equations. In most cases where there are more internal unknown forces in the cables than the number of nodal equilibrium equations, the effect of external load is similar to the case of ordinary reticulated shells and the nodal displacements are due to the elasticity of the members only. Where the number of unknowns is equal to or less than the number of equations, the nodal displacements caused by external load are due to rigid body movements of the net members as well as their elastic properties. A method of analyzing these displacements is proposed. The so‐called fitted load where the nodal displacements are due to the elasticity of the members only is investigated. The behavior of these two type of nets is compared and the implication for the design of cable nets is discussed.

A New Approach for Plate Vibrations: Combination of Transfer Matrix and Finite-Element Technique

When the finite-element method is used in the vibration analysis of plates and shells, it results in large matrices requiring a large digital computer. A commonly used method of reducing the matrix size is to eliminate certain “slave” displacements by minimizing strain energy. The approach requires good judgement in the selection of the “master” displacements and involves additional approximations and some loss of accuracy. In the present method small matrices are obtained without any further approximations and without reducing the number of degrees of freedom. The transfer matrix technique, generally known as the Holzer-Myklestad method, is well known for beams and shafts. The present method is an extension of this idea to plates. The structure is divided into several strips, with a number of nodes on the left and right sections of each strip. Each strip is subdivided into elements and the stiffness and mass matrices are obtained for individual strips. The nodal equilibrium equations are rearranged to obtain a relation between the section variables of the left and the right sections. The section variables are the forces and the displacements of all the nodes on the section. Requirements of displacement continuity and force equilibrium at the nodes, on common sections of two adjacent strips, gives the transfer matrix relation. Successive matrix multiplication finally relates the variables of the left and right boundary of the structure. Boundary conditions require the determinant of a portion of the overall transfer matrix to vanish at the correct frequency. By calculating the determinant at various assumed values of frequency, the correct frequencies are obtained. The method also gives the corresponding mode shapes. The method as applied to several plate problems gives satisfactory results.

A consistent finite element model for the two-dimensional continuum

The finite element approximation to the continuum problem is examined from the viewpoint of the principle of virtual work. It is shown that the usual nodal equilibrium equations for triangular elements are a consistent consequence of a piecewise constant strain field, thus guaranteeing that many results of general continuum theory can be directly applied to the finite element model, and also clarifying the relation between the two models.