Principle Of Virtual Forces Research Articles

Exact structural-property matrices and Timoshenko’s shape functions for nonprismatic frame elements defined by noncentroidal axes

When one is attempting to model framed structures and adopts beam-like finite elements formulated with reference to their centroidal axis—which avoids the interaction between axial and flexural effects—one must, in many practical situations, take into account geometrically complex curved-line finite elements. This fact, which can be quite disadvantageous, is typically called for when solving physically nonlinear problems or modeling nonprismatic beams having cross sections varying asymmetrically around their straight centerlines. This paper proposes then a flexibility-type method based on the principle of virtual forces (PVF) to obtain, according to the Timoshenko beam theory (TBT), exact expressions for the structural properties of nonprismatic frame elements described with reference to noncentroidal axes. The main advantage of this method is that it allows, through the use of straight-line segments, the modeling of complex variable-rigidity frame elements. To interpolate the variable sectional rigidities over the elements, one adopts polynomials of different orders. In addition, this new method allows one to obtain the exact Timoshenko’s shape functions (TSFs) for nonprismatic frame elements with noncentroidal axes. Such shape functions are needed to evaluate deformation-dependent structural properties, such as geometric stiffness. We validate the robustness of the proposed formulation by solving in-plane tapered beams possessing thin-walled monosymmetric cross sections. Using analytical and highly accurate 3D ANSYS responses, this work determines and compares responses described in terms of displacements, rotations, internal forces, and stresses.

Sequential Plastic Method for 2D frames limit analysis

This work focuses on the limit analysis or plastic calculation of slender beam planar frames using a new method. Sufficient static equilibrium equations are proposed by the Principle of Virtual Displacements (PVD) and kinematic compatibility equations are proposed by the Principle of Virtual Forces (PVF). The load level is increased and the structure is solved step by step until the plastic collapse. Equilibrium equations are posed by virtual problems in displacements and compatibility equations are posed by simple virtual problems in equilibrium. The method provides the following results: the collapse load factor, the final collapse mechanism, the bending moments and the accumulated rotations in the plastic hinges in each load step of the structure. This method has advantages over the classical methods; first, with respect to the step-by-step method based on matrix formulation, especially in the case of beams and/or columns with uniform distributed load, and secondly, with respect to the kinematic direct method, since the sequential method provides more information on the quantities involved in the plasticization process of the structure and also goes directly to the calculation of the collapse mechanism without the need to test or combine possible mechanisms.

Nonlinear analysis of semirigid steel frames having nonprismatic shear-deformable members

Starting from the principle of virtual forces (PVF), one devises a flexibility-type method to obtain, based on the Timoshenko beam theory (TBT), exact expressions for the structural properties of nonprismatic frame elements. One adopts polynomials of different orders to interpolate the sectional rigidity values over the elements and employs exact Timoshenko’s shape functions for evaluating deformation-dependent structural properties, such as geometric stiffness. In this respect, we present a mathematically formal process to obtain the changes in the preexisting element forces needed to calculate geometric stiffnesses. The procedure adopted in this paper easily allows one to include higher-order nonlinear terms in the analysis. Robustness of the proposed formulation is observed by conducting geometrically nonlinear analyses of 2D semirigid steel frames possessing nonprismatic members. General nonlinear curves describing the semirigid connections may be easily incorporated into the analysis.

Strain-Gradient Bar-Elastic Substrate Model with Surface-Energy Effect: Virtual-Force Approach.

This paper presents an alternative approach to formulating a rational bar-elastic substrate model with inclusion of small-scale and surface-energy effects. The thermodynamics-based strain gradient model is utilized to account for the small-scale effect (nonlocality) of the bar-bulk material while the Gurtin–Murdoch surface theory is adopted to capture the surface-energy effect. To consider the bar-surrounding substrate interactive mechanism, the Winkler foundation model is called for. The governing differential compatibility equation as well as the consistent end-boundary compatibility conditions are revealed using the virtual force principle and form the core of the model formulation. Within the framework of the virtual force principle, the axial force field serves as the fundamental solution to the governing differential compatibility equation. The problem of a nanowire embedded in an elastic substrate medium is employed as a numerical example to show the accuracy of the proposed bar-elastic substrate model and advantage over its counterpart displacement model. The influences of material nonlocality on both global and local responses are thoroughly discussed in this example.

Time-domain analysis of framed structures based on “exact” structural-property matrices for nonprismatic Timoshenko’s elements

This paper applies a unified process to calculate ”exact” (consistent) finite-element (FE) matrices for framed structures having nonprismatic elements and including shear-deformation and rotational-inertia effects. In this process, the exact expressions for the element stiffness and nodal-load coefficients result from applying the principle of virtual forces (PVF) at the element level. Rigidity values, determined at a certain number of cross sections in the frame element, are employed to describe how the corresponding rigidities vary along its length. For that, interpolation polynomials of different orders are considered. Exact Timoshenko’s shape functions, built under the most general cases of rigidity variation, are used for evaluating the mass matrix coefficients. In the applications, complex 2D frames with nonprismatic elements are considered to simulate bridge structures under seismic excitation and a generic harmonic load. Comparisons with highly accurate response time-histories obtained by employing ANSYS (3D) solid-FE models are effected to verify the robustness of the proposed formulation.

Nonlinear flexibility-based beam element on Winkler-Pasternak foundation

A novel flexibility-based beam-foundation model for inelastic analyses of beams resting on foundation is presented in this paper. To model the deformability of supporting foundation media, the Winkler-Pasternak foundation model is adopted. Following the derivation of basic equations of the problem (strong form), the flexibility-based finite beam-foundation element (weak form) is formulated within the framework of the matrix virtual force principle. Through equilibrated force shape functions, the internal force fields are related to the element force degrees of freedom. Tonti's diagrams are adopted to present both strong and weak forms of the problem. Three numerical simulations are employed to assess validity and to show effectiveness of the proposed flexibility-based beam-foundation model. The first two simulations focus on elastic beam-foundation systems while the last simulation emphasizes on an inelastic beam-foundation system. The influences of the adopted foundation model to represent the underlying foundation medium are also discussed.

Probabilistic multiscale modeling of fracture in heterogeneous materials and structures

The probabilistic aspects of multiscale modeling of the fracture of heterogeneous structures are considered. An approach combining homogenization methods with phenomenological and numerical models of fracture mechanics is proposed to solve the problems of assessing the probabilities of destruction of structurally heterogeneous materials. A model of a generalized heterogeneous structure consisting of heterogeneous materials and regions of different scales containing cracks and crack-like defects is formulated. Linking of scales is carried out using kinematic conditions and multiscale principle of virtual forces. The probability of destruction is formulated as the conditional probability of successive nested fracture events of different scales. Cracks and crack-like defects are considered the main sources of fracture. The distribution of defects is represented in the form of Poisson ensembles. Critical stresses at the tops of cracks are described by the Weibull model. Analytical expressions for the fracture probabilities of multiscale heterogeneous structures with multilevel limit states are obtained. An approach based on a modified Monte Carlo method of statistical modeling is proposed to assess the fracture probabilities taking into account the real morphology of heterogeneous structures. A feature of the proposed method is the use of a three-level fracture scheme with numerical solution of the problems at the micro, meso and macro scales. The main variables are generalized forces of the crack propagation and crack growth resistance. Crack sizes are considered generalized coordinates. To reduce the dimensionality, the problem of fracture mechanics is reformulated into the problem of stability of a heterogeneous structure under load with variations of generalized coordinates and analysis of the virtual work of generalized forces. Expressions for estimating the fracture probabilities using a modified Monte Carlo method for multiscale heterogeneous structures are obtained. The prospects of using the developed approaches to assess the fracture probabilities and address the problems of risk analysis of heterogeneous structures are shown.

A novel strategy to construct exact structural-property matrices for nonprismatic Timoshenko’s frame elements

Assuming Timoshenko’s beam hypothesis, this paper proposes a unified strategy to derive exact finite-element (FE) matrices for framed structures having elements with variable rigidity. Its basic idea is to apply the principle of virtual forces (PVF), at the element level, to obtain a flexibility-based set of equations from which structural-property and nodal-load coefficients can be directly evaluated. The variable physical-geometric characteristics along the frame elements are approximated by polynomials of different orders. For evaluating structural-property coefficients that depend on the deformation of the structure, as e.g. the geometric stiffness coefficients, one employs Timoshenko’s consistent shape functions. A novel process for building them under the most general cases of rigidity variation is presented in this paper. In this study, we particularly apply the technique to effect second-order analyses of 2D frames with nonprismatic elements.

Macro-architectured cellular materials: Properties, characteristic modes, and prediction methods

Macro-architectured cellular (MAC) material is defined as a class of engineered materials having configurable cells of relatively large (i.e., visible) size that can be architecturally designed to achieve various desired material properties. Two types of novel MAC materials, negative Poisson’s ratio material and biomimetic tendon reinforced material, were introduced in this study. To estimate the effective material properties for structural analyses and to optimally design such materials, a set of suitable homogenization methods was developed that provided an effective means for the multiscale modeling of MAC materials. First, a strain-based homogenization method was developed using an approach that separated the strain field into a homogenized strain field and a strain variation field in the local cellular domain superposed on the homogenized strain field. The principle of virtual displacements for the relationship between the strain variation field and the homogenized strain field was then used to condense the strain variation field onto the homogenized strain field. The new method was then extended to a stress-based homogenization process based on the principle of virtual forces and further applied to address the discrete systems represented by the beam or frame structures of the aforementioned MAC materials. The characteristic modes and the stress recovery process used to predict the stress distribution inside the cellular domain and thus determine the material strengths and failures at the local level are also discussed.

Geometric Nonlinear Analysis of Plane Frames With Generically Nonuniform Shear-deformable Members

This paper employs the Direct Stiffness Method (DSM) to carry out geometric nonlinear analysis of plane frames with nonuniform physical-geometric characteristics. At the element level, a flexibility system of equations based on the principle of virtual forces (PVF) is established to calculate the tangent stiffness matrix and the equivalent nodal loads. The formulation allows for the easy modeling of shear-deformable frame elements with generic rigidity variation along their axes. In addition, Green's theorem is considered to express all the necessary section properties in terms of boundary integrals. This considerably simplifies the modeling of complex cross sections of arbitrary shapes. A “boundary-element” mesh is then used to model the geometric description of the cross sections. At the structure level, to determine the nonlinear equilibrium trajectories for the frame, we apply a co-rotational updated Lagrangian formulation along with an incremental-iterative full Newton-Raphson process. Large displacements and internal member forces are accurately reconstituted. Frameworks having elements with geometrically complex cross-sections varying along their axes are analyzed to validate the strategy proposed.

Improved displacement based alternative to force based finite element for nonlinear analysis of framed structures

This paper presents an improved displacement based element (iDBE) for nonlinear finite element analysis of framed structures, which, like the force based elements (FBE), needs neither high order shape functions nor the refinement of the finite element mesh in straight members to attain an accurate solution. To achieve this objective, to the strain fields based on the displacements shape functions of the conventional displacement based element (cDBE) corrective fields are added, which are determined at the element level by the principle of virtual forces. Simple examples involving tapered elements, nonlinear axial-bending interaction and elasto-plastic behavior, are included to illustrate the application of iDBE and compare its performance to cDBE and FBE.

Exact stiffness matrix for nonlocal bars embedded in elastic foundation media: the virtual-force approach

This paper proposes a solution to the exact bar-foundation element that includes the bar nonlocal effect. The exact element stiffness matrix and fixed-end force vector are derived based on the exact element flexibility equation using the so-called natural approach. The virtual-force principle is employed to work out the governing differential compatibility equation as well as the associated end-boundary compatibility conditions. Exact force interpolation functions are used to derive the exact element flexibility equation and can be obtained as the analytical solution of the governing differential compatibility equation of the problem. A numerical example of a nanowire-elastic substrate system is used to verify the accuracy and efficiency of the natural nonlocal bar-foundation model and to demonstrate the superiority over its counterpart, a displacement-based model. The effects of material nonlocality on the system responses are also discussed in the example.

Force‐based derivation of exact stiffness matrix for beams onWinkler‐Pasternak foundation

In this paper, the exact element stiffness matrix for a beam on Winkler‐Pasternak foundation and the fixed‐end force vector due to a linearly distributed load are alternatively derived based on the virtual force principle. The exact element flexibility matrix is at the core of the derivation of the exact element stiffness matrix and is formulated based on the exact force interpolation functions. The virtual force principle is employed to reveal the governing differential compatibility equations as well as the associated end‐boundary compatibility conditions. The exact force interpolation functions of the beam‐foundation system are derived based on the analytical solution of the governing differential compatibility equations of the problem for all combinations of foundation parameters and beam rigidity. This feature is unique to this paper. The matrix virtual force equation is employed to obtain the exact element flexibility matrix using the exact force interpolation functions. The so‐called “natural” element stiffness matrix is obtained by inverting the exact element flexibility matrix. Three numerical examples confirm the accuracy and the efficiency of the natural beam element on Winkler‐Pasternak foundation and show the deficiency of the widely used Winkler foundation model.

Exact Stiffness for Beams on Kerr-Type Foundation: The Virtual Force Approach

This paper alternatively derives the exact element stiffness equation for a beam on Kerr-type foundation. The shear coupling between the individual Winkler-spring components and the peripheral discontinuity at the boundaries between the loaded and the unloaded soil surfaces are taken into account in this proposed model. The element flexibility matrix is derived based on the virtual force principle and forms the core of the exact element stiffness matrix. The sixth-order governing differential compatibility of the problem is revealed using the virtual force principle and solved analytically to obtain the exact force interpolation functions. The matrix virtual force equation is employed to obtain the exact element flexibility matrix based on the exact force interpolation functions. The so-called “natural” element stiffness matrix is obtained by inverting the exact element flexibility matrix. One numerical example is utilized to confirm the accuracy and the efficiency of the proposed beam element on Kerr-type foundation and to show a more realistic distribution of interactive foundation force.

Natural stiffness matrix for beams on Winkler foundation: exact force-based derivation

This paper presents an alternative way to derive the exact element stiffness matrix for a beam on Winkler foundation and the fixed-end force vector due to a linearly distributed load. The element flexibility matrix is derived first and forms the core of the exact element stiffness matrix. The governing differential compatibility of the problem is derived using the virtual force principle and solved to obtain the exact moment interpolation functions. The matrix virtual force equation is employed to obtain the exact element flexibility matrix using the exact moment interpolation functions. The so-called "natural" element stiffness matrix is obtained by inverting the exact element flexibility matrix. Two numerical examples are used to verify the accuracy and the efficiency of the natural beam element on Winkler foundation.

Frame element with lateral deformable supports: Formulations and numerical validation

This paper presents the theory and the numerical validation of three different formulations of nonlinear frame elements with nonlinear lateral deformable supports. The governing differential equations of the problem are derived first and the three different finite element formulations are then presented. The first model follows a displacement-based formulation, which is based on the virtual displacement principle. The second one follows the force-based formulation, which is based on the virtual force principle. The third model follows the Hellinger–Reissner mixed formulation, which is based on the two-field mixed variational principle. The selection of the displacement and force interpolation functions for the different formulations is discussed. Tonti’s diagrams are used to conveniently represent the equations governing both the strong and the weak forms of the problem. The general matrix equations of the three formulations are presented, with some details on the issues regarding the elements’ implementations in a general-purpose finite element program. The convergence, accuracy, and computational times of the three elements are studied through a numerical example. The distinctive element characteristics in terms of force and deformation discontinuities between adjacent elements are discussed. The capability of the proposed frame models to trace the softening response due to softening of the foundation is also investigated. Overall, the force-based and the mixed models are much more accurate than the displacement-based model and require very few elements to reach the converged solution. The force-based element is slightly more accurate than the mixed model, but it is more prone to numerical instabilities as it involves inverting the element flexibility matrix.

Collapse Analysis: Large Inelastic Deformations Analysis of Planar Frames

In an attempt to trace the collapse of structures in seismic events, this paper discusses an alternative approach to the formulation and solution of the large deformation inelastic problem in planar frame structures. A beam–column element that includes the effect of geometric and material nonlinearities is developed using the flexibility approach. The formulation uses force interpolation functions and the principle of virtual forces in rate form. It is formulated in the context of the state space approach, where the global system of conservation equations and the local constitutive equations are solved simultaneously. The differential-algebraic system solver is used to solve the resulting system of equations. Numerical examples are presented to illustrate the functionality of the analysis using the newly formulated approach.

Mixed methods and flexibility approaches for nonlinear frame analysis

The principle of virtual forces has long been an attractive tool for linear matrix structural analysis because it provides a means to compute the exact flexibility matrix for a linear, elastic, non-prismatic Bernoulli–Euler beam element while the classical stiffness method, based on the principle of virtual displacements, does not. Nonlinear flexibility methods have recently appeared in the literature as a means of improving the accuracy of frame analysis when the curvature fields are more complex than the moment fields (e.g., when inelastic response is important). This paper surveys the basic methods of formulating nonlinear Bernoulli–Euler beam elements based on classical and mixed variational principles. We show that the variational structure provided by the Hellinger–Reissner and Hu–Washizu functionals gives a framework that reaps the benefits of the so-called nonlinear flexibility methods while allowing a standard treatment of non-linear problems by Newton's method.

DYNAMIC SHAPE CONTROL OF SOLIDS AND STRUCTURES BY THERMAL EXPANSION STRAINS

The present article is concerned with the dynamic shape control of solids and structures by means of thermal expansion strains or, equivalently, by means of thermal expansion stresses. We study transient disturbances produced by imposed forces, and we wish to identify a transient temperature distribution that, when superimposed, leads to zero total displacements of the body. The derivation is based on the theorem of work expended and on Graffi's theorem. Using the anisotropic constitutive equations of linear thermoelasticity, we arrive at a dynamic extension of the principle of virtual forces and at a dynamic extension of Maysel's formula of thermoelasticity. Comparing these two convolution statements, it is found that, in order to make the total displacement zero everywhere in the body, the thermal expansion stress in the thermal problem should be the quasi-static stress minus the thermal expansion stress in the force problem. Any quasi-static stress due to a thermal loading may be added to the thermal expansion stress without changing the validity of this theorem. The practical application of this result is facilitated when the production of heat due to deformation may be neglected and when the applied forces are separable in space and time. The validity of the analytical solution is checked, by means of finite element computations.

THE PROBLEMS OF LIMIT LOAD ANALYSIS AND OPTIMIZATION USING EQUILIBRIUM FINITE ELEMENTS

The equilibrium dual discrete mathematical models of the problems of limit load analysis and optimization are investigated in the article. These models are presented in terms of static and kinematic formulation using equilibrium finite elements. In these mathematical models the possible discontinuities of displacement velocities are evaluated and the velocity of energy dissipation is estimated not only within the volume of finite elements, but at the plastic surfaces between elements. At first, on the basis of the energy principle of the maximum external power [1,2] the general mathematical models (3) and (7) of static formulation of limit load analysis and optimization problems are created. In these models the yield conditions are controlled not only within the volume, but also at the surfaces of finite elements. The equilibrium finite elements and interpolation functions of strains (9) are used for discretization of these models. The constancy of external power is taken as the optimum criterion. The discrete expressions of fundamental relationships—equilibrium and geometric equations, yield conditions (10)-(12) for finite element and (14)-(17) for the discrete model of a body are developed. The discrete expressions of yield conditions are given using the classic collocation methods: collocation at the point, collocation at the sphere (element) and Bubnov-Galiorkin's collocation method [11,12]. The equilibrium equations of discrete structure are developed on the basis of virtual displacement principle while geometrical equations are derived using virtual force principle. In contrast to the approach of other authors, yield conditions and geometrical equations are described not only within finite elements, but also at the surfaces between elements. That helps to design the dual discrete mathematical models of the problems (21)-(26), in which the discontinuities of displacement velocities and the velocity of energy dissipation in the place of those discontinuities are estimated. The mathematical models (22), (24), (26), (29) and (32) of kinematic problem formulation are developed from sensible static formulations by Lagrange's multiplier method. The modified mathematical models (27)-(29) are presented. In these models the equilibrium equations are eliminated or the geometrical equations are transformed into compatible equations of plastic stress velocities, in this way decreasing the number of equations and unknown values. The dependence of the numerical results (limit load) of the frame on the approximation degree of bending moments, as well as on the discretization method of yield conditions are illustrated. In table 1 the values of limit loading parameter F 0 and their error of calculation ΔF 0 (per cent, in comparison with the analytic solution F 0 = 0,4662M 0) are presented. They are given for the first and second order finite elements with linear and parabolic distribution of bending moments using different discrete yield conditions and a different number of finite elements. The numerical result shows, that the discretization of yield conditions by Bubnov-Galiorkin's method gives the best accuracy and stable solutions. By discretizing yield conditions using the point's collocation and collocation at the element, the accurancy of numerical results depends not only on the number of elements, but also on a more or less successful choice of finite elements net.