Equivalent Load Vector Research Articles

Determination of influence lines for structural responses with uncertainties in properties of the structure

Classical ways of determination of influence lines for structural responses are often inefficient due to high computational cost, especially if properties of the structure are uncertain. The combination of Muller-Breslau principle and the finite element analysis can resolve this problem. In this study, formulations for new finite elements tailored for use in determination of influence lines for structural responses are proposed. They are then used in the problem of uncertainty propagation to obtain uncertain ordinates of the influence lines. The expressions for the element stiffness matrix and equivalent nodal load vector are derived consistently from the proposed displacement field of the element. The proposed finite elements can be adapted directly in existing finite element packages since they do not need remeshing. Studied examples show the correctness and the efficiency of the proposed method. From the results of uncertainty propagation problem, large uncertainties in the ordinates of influence lines for structural responses due to small uncertainties in structural properties should be aware of.

Simulation and Analysis of the Temperature Field and the Thermal Stress of an Inverted-Siphon Concrete Structure Based on the Contact Friction Element

Mass concrete structures are prone to suffer temperature destruction during construction and operation because of their own mechanical properties and the role of cement hydration; therefore, the law of temperature field and temperature stress changes is crucial. This paper is based on the heat conduction theory and contact friction element theory under general external load. The heat conduction matrix, node thermal load vector, equivalent stiffness-constraint matrix and equivalent load vector of the contact friction element are each added to the whole heat conduction matrix, thermal load vector and total stiffness matrix, and the total load vector according to the finite element integration rule to establish the thermal-stress calculation model of the contact friction element. Combined with the Qi River inverted siphon project, this paper analyzed the temperature field and temperature stress using the three-dimensional finite element method, considering the changes of concrete thermodynamic parameters with age and the outside air temperature and other conditions, and obtained the distribution and variation in temperature field and temperature stress during the construction of an inverted siphon. The research results are of great significance for temperature control and crack prevention of inverted siphon structures. The thermal-stress calculation model of the contact friction element overcomes the limitations that it is difficult to use the current contact surface model to perform the temperature field simulation and further improves the application of the contact friction element model in the simulation of the temperature field and the temperature stress of mass concrete.

Finite element for the dynamic analysis of pipes subjected to water hammer

An axisymmetric finite element is developed for the dynamic analysis of pipes subjected to water hammer. The analysis seamlessly captures the pipe response due to water hammer by solving first the water hammer mass and momentum conservation equations, to recover the spatial–temporal distribution of the internal pressure, and then uses the predicted pressure history to form a time-dependent energy equivalent load vector within a finite element model. The study then determines the pipe response by solving the finite element discretized equations of motion. The formulation of the pipe response is based on Hamilton’s principle in conjunction with a thin shell theory formulation, and captures inertial and damping effects. The results predicted by the model are shown to be in agreement with those based on an axisymmetric shell model in Abaqus for static, free vibration and transient responses for benchmark problems. The water hammer and structural models are seamlessly integrated to enable advancing the transient solution well into the time domain to capture the effect of reflected pressure wave due to water hammer. The results indicate that the radial stress/displacement oscillations approach those of the quasi-steady response when the valve closure time exceeds eight times the period of radial vibrations obtained for the case of instantaneous valve closure.

Advanced 3-D beam element including warping and distortional effects for the analysis of spatial framed structures

In this paper an advanced 32 × 32 stiffness matrix and the corresponding nodal load vector of a 3-D beam element of arbitrary cross section taking into account shear deformation, generalized warping (shear lag effects) and distortional effects due to both flexure and torsion is presented. Nonuniform distortional/warping distributions are taken into account by employing 10 additional degrees of freedom per node. Local stiffness matrix and local equivalent nodal load vector of the element are computed numerically applying the Analog Equation Method (AEM), a Boundary Element Method (BEM) based technique. Warping and distortional functions as well as geometric constants of the cross section are evaluated employing a 2-D BEM approach. The developed element is incorporated into a standard Direct Stiffness Method (DSM) algorithm employed for the solution of spatial frames. The problem of handling distortional/warping transmission conditions in non-aligned members is alleviated by using an approximate technique for a proper transformation of higher-order degrees of freedom at frame joints.

Efficient structure topology optimization by using the multiscale finite element method

Efficient SIMP and level set based topology optimization schemes are proposed based on the computation framework of the multiscale finite element method (MsFEM). In the proposed optimization schemes, the equilibrium equations are solved on a coarse-scale mesh and the design variables are updated on a fine-scale mesh. To describe more complex deformation, a multi-node coarse element is also presented in the MsFEM computation. In the MsFEM, a multiscale shape function is constructed numerically and employed to obtain the equivalent stiffness matrix and load vector of the multi-node coarse element. In the optimization schemes with the MsFEM, the coarse elements are divided into two categories: homogeneous and heterogeneous. For the homogeneous coarse elements, their multiscale shape functions are constructed only once before the iterations. Since the material distribution is varying locally in most of the iterations, one only needs to reconstruct them of a small part of the coarse elements where the material distribution is changed by comparison with that in the previous iteration step. This will save lots of computational cost. In addition, due to the independence of each coarse element, the constructions of the multiscale shape functions could be easily proceeded in parallel. In this work, the computational accuracy and efficiency of this method is investigated in detail, as well as the speedup ratio and parallel efficiency when using multiple processors to construct the multiscale shape functions simultaneously. Furthermore, several 2D and 3D examples show the effectiveness and efficiency of the proposed optimization schemes based on the MsFEM analysis framework.

A static analysis of three-dimensional sandwich beam structures by hierarchical finite elements modelling

A static analysis of three-dimensional sandwich beam structures using one-dimensional modelling approach is presented within this paper. A family of several one-dimensional beam elements is obtained by hierarchically expanding the displacements over the cross-section and letting the expansion order a free parameter. The finite element approximation order over the beam axis is also a formulation free parameter (linear, quadratic and cubic elements are considered). The principle of virtual displacements is used to obtain the problem weak form and derive the beam stiffness matrix and equivalent load vectors in a nuclear, generic form. Displacements and stresses are presented for different load and constraint configurations. Results are validated towards three-dimensional finite element solutions and experimental results. Sandwich beams present a three-dimensional stress state and higher-order models are necessary for an accurate description. Numerical investigations show that fairly good results with reduced computational costs can be obtained by the proposed finite element formulation.

Numerical and Experimental Investigation on the Structural Behaviour of a Horizontal Stabilizer under Critical Aerodynamic Loading Conditions

The aim of the proposed research activity is to investigate the mechanical behaviour of a part of aerospace horizontal stabilizer, made of composite materials and undergoing static loads. The prototype design and manufacturing phases have been carried out in the framework of this research activity. The structural components of such stabilizer are made of composite sandwich panels (HTA 5131/RTM 6) with honeycomb core (HRH-10-1/8-4.0); the sandwich skins have been made by means of Resin Transfer Moulding (RTM) process. In order to assess the mechanical strength of this stabilizer, experimental tests have been performed. In particular, the most critical inflight recorded aerodynamic load has been experimentally reproduced and applied on the stabilizer. A numerical model, based on the Finite Element Method (FEM) and aimed at reducing the experimental effort, has been preliminarily developed to calibrate amplitude, direction, and distribution of an equivalent and simpler load vector to be used in the experimental test. The FEM analysis, performed by using NASTRAN code, has allowed modelling the skins of the composite sandwich plates by definition of material properties and stack orientation of each lamina, while the honeycomb core has been modelled by using an equivalent orthotropic plate. Numerical and experimental results have been compared and a good agreement has been achieved.

A new equivalent friction element for analysis of cable supported structures

An equivalent friction element is proposed to simulate the friction in cable-strut joints. Equivalent stiffness matrixes and load vectors of the friction element are derived and are unified into patterns for FEM by defining a virtual node specially to store internal forces. Three approaches are described to verify the rationality of the new equivalent friction element: applying the new element in a cable-roller model, and numerical solutions match well with experimental results; applying the element in a continuous sliding cable model, and theoretical values, numerical and experimental results are compared; and the last is applying it in truss string structures, whose results indicate that there would be a great error if the cable of cable supported structures is simulated with discontinuous cable model which is usually adopted in traditional finite element analysis, and that the prestress loss resulted from the friction in cable-strut joints would have adverse effect on the mechanical performance of cable supported structures.

Uncertainty Analysis of Static Plane Problems by Intervals

We present a new interval-based formulation for the static analysis of plane stress/strain problems with uncertain parameters in load, material and geometry. We exploit the Interval Finite Element Method (IFEM) to model uncertainties in the system. Overestimation due to dependency among interval variables is reduced using a new decomposition strategy for the structural stiffness matrix and the nodal equivalent load vector. Primary and derived quantities follow from minimization of the total energy and they are solved simultaneously and with the same accuracy by means of Lagrangian multipliers. Two different element assembly strategies are introduced in the formulation: one is Element-by-Element, and the other resembles conventional assembly. In addition, we implement a new variant of the interval iterative enclosure method to obtain outer and inner solutions. Numerical examples show that the proposed interval approach guarantees to enclose the exact system response.

Study on bending of hard core sandwich plates based on Reissner’s sandwich theory

In the traditional theories of sandwich plates the core is soft so the in-plane stress components and bending stiffness are neglected. This paper explores the bending problem of hard core sandwich plates based on the revised Reissner's hypothesis. The basic equations and boundary conditions expressed with three generalized displacements are established and simplified according to the principle of minimum potential energy. In addition this paper presents the finite element model for the bending problem of hard core sandwich plates. Based on this model a four-node quadrilateral isoparametric element is constructed, the element stiffness matrix and equivalent nodal load vector are derived. It is built the solid model for ANSYS. According to the numerical analysis the finite element solutions converge to the analytical solutions with sufficient accuracy and also consistent with solutions of ANSYS.

General exact harmonic analysis of in-plane timoshenko beam structures

The exact solution for the problem of damped, steady state response, of in-plane Timoshenko frames subjected to harmonically time varying external forces is here described. The solution is obtained by using the classical dynamic stiffness matrix (DSM), which is non-linear and transcendental in respect to the excitation frequency, and by performing the harmonic analysis using the Laplace transform. As an original contribution, the partial differential coupled governing equations, combining displacements and forces, are directly subjected to Laplace transforms, leading to the member DSM and to the equivalent load vector formulations. Additionally, the members may have rigid bodies attached at any of their ends where, optionally, internal forces can be released. The member matrices are then used to establish the global matrices that represent the dynamic equilibrium of the overall framed structure, preserving close similarity to the finite element method. Several application examples prove the certainty of the proposed method by comparing the model results with the ones available in the literature or with finite element analyses.

Dynamic Structural Analysis of Large Dams by Fourier SEM

Spectral element method (SEM), which combines the ideas of the finite element method (FEM) with the theory of spectral method, is being in the initial stage of developing for the static and dynamic analysis of large dams. The best advantage of SEM is that it can arrive at so-called spectral accuracy out of FEMs reach. In this paper, the Fourier SEM has been first used in dynamic analysis of large dams in order to improve the accuracy and efficiency of numerical results and procedure. The study begins with the governing equation of motion of large dams, then deduces the corresponding SEM stiffness, mass (damping) matrix and equivalent load vector taking advantage of the Fourier interpolation polynomials to approximate the unknowns in spatial domains. This paper also reveals the valuable application of SEM in complicated structural engineering. The formulation proposed in this paper can also be applied to the general dynamic analysis of physical structures.

Dynamic Analysis of Large Dams in Frequency-Domain Based on Fourier SEM

In this paper, we applyFourierSpectral Element Method(SEM) to the dynamic analysis of large dams, and take advantage of its high numerical accuracy and exponential rate of convergence and excellent geometrical adaptability to improve the computational procedure, efficiency and results of dynamic problem of large dams in numerical calculation. Based on the governing dynamic equation of large dams with the boundary nonlinearity, this paper derives theFourierspectral stiffness, mass, damping matrix and equivalent load vector in terms of discreteFourierspectral series, so a spectral formulation of dynamic analysis of large dams can be reached. For the purpose of higher numerical efficiency,Fast Fourier Transform(FFT) is also adopted in this paper. AlthoughSEMhas few applications in the analysis of solid structures, this paper indicates thatSEMcould be used in solid construction such as large dams as good as other preferable fields.

Modeling of Axially Loaded Nanowires Embedded in Elastic Substrate Media with Inclusion of Nonlocal and Surface Effects

Nonlocal and surface effects are incorporated into a bar‐elastic substrate element to account for small‐scale and size‐dependent effects on axial responses of nanowires embedded in elastic substrate media. The virtual displacement principle, employed to consistently derive the governing differential equation as well as the boundary conditions, forms the core of the displacement‐based finite element formulation of the nanowire‐elastic substrate element. The element displacement shape functions, analytically derived based on homogeneous solution to the governing differential equilibrium equation of the problem, result in the exact element stiffness matrix and equivalent load vector. Two numerical simulations employing the proposed model are performed to study characteristics and behavior of the nanowire‐substrate system. The first simulation involves investigation of responses of the wire embedded in elastic substrate. The second examines influences of several system parameters on the contact stiffness and reveals the size‐dependent effect on the effective Young′s modulus of the system.

Non-Dimensional Vibration of Timoshenko Beams with Axial Loads

The paper is supposed to establish a non-dimensional approach to analyze the free vibration of axially loaded beams with different kinds of boundary conditions, based on the Timoshenko’s beam theory and the method of separation of variables. When the mode-shapes and frequencies are known, it is convenient to get the equivalent load vector, equivalent mass matrix and the frequencies matrix, then, the characteristics of vibration of axially loaded beams will be obtained by the approach. The results and conclusions were also helpful to the anti-blast analysis of columns

Incremental Launching Construction Analysis of Continuous Skewed Bridge with Uniform Cross Section

The finite element formulation was used to analyze the influences of structural parameters such as the skewed angle and the stiffness ratio of flexure to torsion on the distributions of moment and torsion. The moment and torsion factors figure were presented according to the different structural parameters such as the skewed angle and the stiffness ratio of flexure to torsion. Some equivalent load vectors of non-nodal load acting on the skewed supported beam were derived.

Matrix formulation of dynamic design of structures with semi-rigid connections

The structures with semi-rigid connections comprise systems with the connections in joints which are not absolutely rigid, but allow, in general, some relative movements in directions of generalized displacements. Such type of connections is considered very little, or not at all, in designing of structures in today's engineering practice. If the influence of rigidity of semi-rigid connections is underestimated, and they are treated in the design as pinned, it has a negative impact on cost of a structure. But if it is overstated, the calculation results are not on the side of safety, what is reflected on bearing capacity, durability and stability, especially in the case of precast structures. Therefore Eurocodes take due account to the structures with semi-rigid connections. Matrix formulation of the analysis of systems with semi-rigid connections opens wide possibilities for relatively easy calculation by use of computers that is shown by example of seismic design. The interpolation functions, stiffness matrix, equivalent load vectors, and the consistent mass matrix are presented in this paper, particularly with an emphasis on systems with semi-rigid connection.

Structural matrices of a curved-beam element

This article presents the differential system that governs the mechanical behaviour of a curved-beam element, with varying cross-section area, subjected to generalized load. This system is solved by an exact procedure or by the application of a new numerical recurrence scheme relating the internal forces and displacements at the two end-points of an increase in its centroid-line. This solution has a transfer matrix structure. Both the stiffness matrix and the equivalent load vector are obtained arranging the transfer matrix. New structural matrices have been defined, which permit to determine directly the unknown values of internal forces and displacements at the two supported ends of the curved-beam element. Examples are included for verification.

Finite Elements on Generalized Elastic Foundation in Timoshenko Beam Theory

Using the Vlasov foundation model, a modified approach of the continuous beam on elastic supports, leading to both a mechanical model and the proper foundation parameters of the generalized foundation is shown. Two formulations of the beam finite-element with shear deformation effect, resting on a two-parameter elastic foundation, characterized by distinct contributions of normal and rotary reactions are presented. The behavior of the second foundation parameter in the two formulations is governed by the bending cross section rotation of a beam. The first formulation, yielding a free-of-meshing stiffness matrix and equivalent nodal load vector, is based on the transcendental or “exact” solution of the governing differential equation of the beam resting on the elastic layer of constant thickness. Considering a linear variation of the layer thickness along the beam, the second formulation is based on the assumed polynomial displacement field. Numerical comparisons with the exact approach show that the cubic formulation leads to better results when the foundation parameters are variables. The practical utility of the analogy between a tensile axial force and the second foundation parameter is exemplified, too.

Accurate Stiffness Matrix for Nonprismatic Members

In this note, the transfer matrix method is adopted to deduce general expressions for the components of the stiffness matrix and equivalent node load vector of nonprismatic members, and the effect of warping is not considered. State vectors are introduced to describe the nodal forces and displacements of a structural member. The relation between the state vectors of the left node and the right node of the member is given by a matrix referred to as the transfer matrix. It is found that the stiffness matrix of the member can be expressed in terms of the transfer matrix. Therefore, an accurate expression for the stiffness matrix can be obtained as long as the corresponding transfer matrix can be accurately determined. The method proposed is a general procedure for the stiffness matrix derivation of both continuous nonprismatic members and discontinuous nonprismatic members. The correctness of the obtained stiffness expressions is verified by two simple numerical examples.