Bending Of Elastic Beams Research Articles

Fast and Compact Partial Differential Equation (PDE)-Based Dynamic Reconstruction of Extended Position-Based Dynamics (XPBD) Deformation Simulation

Dynamic simulation is widely applied in the real-time and realistic physical simulation field. How to achieve natural dynamic simulation results in real-time with small data sizes is an important and long-standing topic. In this paper, we propose a dynamic reconstruction and interpolation method grounded in physical principles for simulating dynamic deformations. This method replaces the deformation forces of the widely used eXtended Position-Based Dynamics (XPBD), which are traditionally derived from the gradient of the energy potential defined by the constraint function, with the elastic beam bending forces to more accurately represent the underlying deformation physics. By doing so, it establishes a mathematical model based on dynamic partial differential equations (PDE) for reconstruction, which are the differential equations involving both the parametric variable u and the time variable t. This model also considers the inertia forces caused by acceleration. The analytical solution to this model is then integrated with the XPBD framework, built upon Newton’s equations of motion. This integration reduces the number of design variables and data sizes, enhances simulation efficiency, achieves good reconstruction accuracy, and makes deformation simulation more capable. The experiment carried out in this paper demonstrates that deformed shapes at about half of the keyframes simulated by XPBD can be reconstructed by the proposed PDE-based dynamic reconstruction algorithm quickly and accurately with a compact and analytical representation, which outperforms static B-spline-based representation and interpolation, greatly shortens the XPBD simulation time, and represents deformed shapes with much smaller data sizes while maintaining good accuracy. Furthermore, the proposed PDE-based dynamic reconstruction algorithm can generate continuous deformation shapes, which cannot be generated by XPBD, to raise the capacity of deformation simulation.

Improving Realism of Facial Interpolation and Blendshapes with Analytical Partial Differential Equation-Represented Physics

How to create realistic shapes by interpolating two known shapes for facial blendshapes has not been investigated in the existing literature. In this paper, we propose a physics-based mathematical model and its analytical solutions to obtain more realistic facial shape changes. To this end, we first introduce the internal force of elastic beam bending into the equation of motion and integrate it with the constraints of two known shapes to develop the physics-based mathematical model represented with dynamic partial differential equations (PDEs). Second, we propose a unified mathematical expression of the external force represented with linear and various nonlinear time-dependent Fourier series, introduce it into the mathematical model to create linear and various nonlinear dynamic deformations of the curves defining a human face model, and derive analytical solutions of the mathematical model. Third, we evaluate the realism of the obtained analytical solutions in interpolating two known shapes to create new shape changes by comparing the shape changes calculated with the obtained analytical solutions and geometric linear interpolation to the ground-truth shape changes and conclude that among linear and various nonlinear PDE-based analytical solutions named as linear, quadratic, and cubic PDE-based interpolation, quadratic PDE-based interpolation creates the most realistic shape changes, which are more realistic than those obtained with the geometric linear interpolation. Finally, we use the quadratic PDE-based interpolation to develop a facial blendshape method and demonstrate that the proposed approach is more efficient than numerical physics-based facial blendshapes.

A corrected finite-difference scheme for the flexure equation with abrupt changes in coefficient

Abstract. The fourth-order differential equation describing elastic flexure of the lithosphere is one of the cornerstones of geodynamics that is key to understanding topography, gravity, glacial isostatic rebound, foreland basin evolution, and a host of other phenomena. Despite being fully formulated in the 1940s, a number of significant issues concerning the basic equation have remained overlooked to this day. We first explain the different fundamental forms the equation can take and their difference in meaning and solution procedures. We then show how numerical solutions to flexure problems as they are currently formulated are in general potentially unreliable in an unpredictable manner for cases in which the coefficient of rigidity varies in space due to variations of the elastic thickness parameter. This is due to fundamental issues related to the numerical discretisation scheme employed. We demonstrate an alternative discretisation that is stable and accurate across the broadest conceivable range of conditions and variations of elastic thickness, and we show how such a scheme can simulate conditions up to and including a completely broken lithosphere more usually modelled as an end-loaded, single, continuous plate. Importantly, our scheme will allow breaks in plate interiors, allowing, for instance, the creation of separate blocks of lithosphere which can also share the support of loads. The scheme we use has been known for many years but remains rarely applied or discussed. We show that it is generally the most suitable finite-difference discretisation of fourth-order, elliptic equations of the kind describing many phenomena in elasticity, including the problem of bending of elastic beams. We compare the earlier discretisation scheme to the new one in one-dimensional form and also give the two-dimensional discretisation based on the new scheme. We also describe a general issue concerning the numerical stability of any second-order finite-difference discretisation of a fourth-order differential equation like that describing flexure wherein contrasting magnitudes of coefficients of different summed terms lead to round-off problems, which in turn destroy matrix positivity. We explain the use of 128 bit floating-point storage for variables to mitigate this issue.

Structural performance of composite WikiHouse beams from CNC-cut timber panels

Computer-numerical-control (CNC) fabrication of interlocking-plate timber structures is a promising form of construction for housing with the potential to be socially, economically and environmentally sustainable. The primary mechanisms of load transfer in these structures rely on direct contact and friction between interlocking elements, without the nails and screws used in conventional timber structures. The development of these connections is relatively new, and therefore the application of interlocking plates structural systems in real projects is so far limited. In this study, the WikiHouse interlocking-plate timber structural system for digitally fabricated houses is presented from a design and fabrication point of view. The main structural elements of the system, the beams and columns, are hollow section members fabricated using computer-numerical-control (CNC) cut plywood panels. In the first part of the paper, the concept of Wikihouse and its fabrication process are presented. Then, the performance of the structural beams is investigated by means of experimental testing on full scale 5 m specimens. Finally, an analytical model to calculate the beam capacity and displacements is derived based on elastic beam bending and joint flexibility. Results show that the specimens failed in a ductile manner. Furthermore, it was found that the joints between the panels introduce extra flexibility to element, and that rigid-body rotations occurring in the joints within the span make a substantial contribution to the overall deflection.

Bending a gel beam with consideration of entanglements and finite extensibility

A hybrid beam containing a gel layer attached on the top of an elastic non-swellable substrate has been normally used to manufacture diverse sensors and actuators. Generally, when the system uses different gels, it is necessary to develop differentiated models for the hybrid beam. Recently, on the basis of the generalized ideal elastomeric gel model, a unified relationship between the gels' swelling and the elastic beam's bending curvature is formulated by Cai, which is separated from some particular swelling mechanisms of gels. He plugged neo-Hookean model and Gent model of the stretching polymer network to the generalized equations. However, the polymer networks usually have many entanglements due to the uncrossability of the network chains as well as a finite extensibility resulting from the full stretching of the network chains. The effect of entanglements cannot be captured by these two models. In this letter, the nonaffine model proposed by Davidson and Goulbourne is adopted to the generalized equations derived by Cai. The results show that entanglements as well as chain extensibility have important influences on the mechanical bending behaviors of the gels' beam.

Diffusion-induced bending of viscoelastic beams

The contribution of local volumetric change due to the diffusion/migration of solute atoms to viscoelastic deformation is incorporated in the theory of linear viscoelasticity, following the elastic theory of diffusion-induced stress. Three-dimensional constitutive relationship in differential form for diffusion-induced stress in linear viscoelastic materials is proposed. Using the correspondence principle between linear viscoelasticity and linear elasticity and the results from the diffusion-induced bending of elastic beams, the radii of curvature of the centroidal plane of viscoelastic beams of single layer and bilayer with top layer being viscoelastic in the transform domain are obtained. For viscoelastic beams of single layer, closed-form solution of the radius of curvature of the centroidal plane is derived, and the radius of curvature is inversely proportional to the diffusion moment created by non-uniform distribution of solute atoms. For the condition of constant concentration on free surface, there is overshoot behavior; for the condition of constant flux on free surface, there is no overshoot behavior. For viscoelastic beams of bilayer with top layer being the Maxwell-type standard material, the numerical results show the presence of the overshoot behavior for a very compliant elastic layer under the condition of constant concentration on free surface, and there is no overshoot behavior under the condition of constant flux on free surface.

A simple and effective nonlinear elastic one-dimensional model for the structural analysis of masonry arches

In this paper the static response of a masonry arch is studied by way of a one-dimensional nonlinear elastic model in which masonry is regarded as a material with bounded tensile and compressive strengths. By following an approach analogous to that followed in the theory of bending of elastic beams, the equilibrium problem for the arch leads to a free-boundary, nonlinear differential problem. An approximate solution to such problem can be pursued by means of an ad hoc iterative procedure, illustrated in detail herein. The results obtained in three case studies are compared with some numerical and experimental results available in the literature. In addition, the case of an actual arch undergoing spreading of the springings is considered, and the distribution and possible evolution of the cracking pattern discussed.

Scaling laws for displacement of elastic beam by energy method

This paper introduces the Engesser's first theorem to the area of structural similarity. General derivation of the scaling laws is described and applied to problem of symmetrical bending of elastic beam under simultaneous application of various loading types. The applied loads considered are concentrated forces, distributed forces, and self-weight load. New sets of complete similarity conditions and scaling laws are established, especially for the problem of nonlinearly elastic beams. It has been shown that the complete similarity condition can be achieved by a geometrically similar model or distorted model satisfying an additional condition. The scaling laws are verified by the problems of a nonlinearly elastic cantilever beam and linearly elastic 2-D frame, which the displacement solutions are determined by analytical and numerical approaches, respectively. These scaling laws are found to accurately predict the displacement of the prototypes in all cases.

Analytical solution for the elastic bending of beams lying on a linearly variable Winkler support

The present paper considers the analytical static bending response of a finite uniform free-free Euler-Bernoulli elastic beam resting on a Winkler elastic foundation with a spatially inhomogeneous stiffness coefficient. Specifically, a linear variation of the foundation coefficient is assumed. The beam-foundation system is loaded by a force and a moment applied at one beam's (top) end, a configuration which may resemble that of a foundation pile. The analytical solution of the governing Ordinary Differential Equation is derived and represented in explicit closed-form in terms of generalized hypergeometric functions. Through the derived solution, parametric analyses are carried out, by interpreting the parametric variation of the mechanical response of the beam-foundation system due to changes in its mechanical properties.

Analytical solution for the elastic bending of beams lying on a variable Winkler support

The interaction between structures and supporting media is crucial for several engineering applications. Among existing models, the elastic one originally attributed to Emil Winkler is rather well-known to both researchers and engineers, with main reference to a constant foundation modulus. The present paper reviews first the state of the art on this model and then considers, analytically, the little-explored case of a space-dependent stiffness coefficient. The analytical solution of the ordinary differential equation describing the static deflection of a simply-supported Euler–Bernoulli elastic beam lying on a variable Winkler elastic foundation is sought. Specifically, the case of a nonlinear, minus four power variation of the stiffness coefficient is considered, allowing for closed-form representations. These are derived and examined in view of interpreting their parametric variations due to changes in the mechanical properties of the beam-foundation system. In the end, a consistent validation comparison between analytical solution and alternative reimplemented numerical treatments is produced.

Design of a Cantilever - Type Rotating Bending Fatigue Testing Machine

This research is centered on the design of a low–cost cantilever loading rotating bending fatigue testing machine using locally sourced materials. The design principle was based on the adaptation of the technical theory of bending of elastic beams. Design drawings were produced and components/materials selections were based on functionality, durability, cost and local availability. The major parts of the machine: the machine main frame, the rotating shaft, the bearing and the bearing housing, the specimen clamping system, pulleys, speed counter, electric motor, and dead weights; were fabricated and then assembled following the design specifications. The machine performance was evaluated using test specimens which were machined in conformity with standard procedures. It was observed that the machine has the potentials of generating reliable bending stress – number of cycles data; and the cost of design (171,000 Naira) was lower in comparison to that of rotating bending machines from abroad. Also the machine has the advantages of ease of operation and maintenance, and is safe for use.

Boundary conditions for elastic beam bending

For beam bending problem, the reciprocal theorem and P–N solution are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order. Through generalizing the method proposed by Gregory and Wan, a set of necessary conditions on the edge-data for the existence of a rapidly decaying solution is established. When stress and mixed conditions are imposed on the beam edge, these decaying state conditions are derived explicitly, and they are used for the correct formulation of boundary conditions for the interior solution. For the stress data, our boundary conditions coincide with those obtained in conventional forms of beam theories. More importantly, the appropriate boundary conditions with two different sets of mixed edge-data are obtained for the first time. To cite this article: Y. Gao et al., C. R. Mecanique 335 (2007).

Robustness of the sheet metal springback cup test

The robustness of a proposed test for elastic springback characterization of sheet metal has been examined using a matrix of defined experimental errors. A series of flat bottom deep drawn cups made from AISI 1010 steel sheet were examined. It was found that misalignment of the blank over the forming tool and error in the vertical location where the springback ring was cut from the cup sidewall had the largest effect on the resulting springback opening. Other experimental errors involving cup height and ring width were found to be less important. The effect of in-plane anisotropy of mechanical properties on springback was negligible. The results are examined in terms of measured through thickness residual stresses and elastic bending of beams with circumferential thickness gradients.

Effect of Loading on Stress Intensification Factors

The basic objective of this investigation is to determine the effect of loading on the stress intensification factors of Markl’s fatigue evaluation method for metal piping. Markl’s method is based on the fatigue testing of 4 in. schedule 40 carbon steel cantilever piping. Subsequent testing using a four-point loading showed that the S-N data were different from that predicted by the procedure and equation developed by Markl. Markl’s method is based on determining the elastic-plastic forces in a piping system by multiplying the elastic system stiffness by the actual deflection. In this manner a fictitious force is calculated to determine piping stresses assuming the elastic beam bending equation, Mc/I, applies even in partially plastic pipes. Previous analytical work on this topic by Rodabaugh and Scavuzzo (“Fatigue of Butt Welded Pipe and the Effect of Testing Methods–Report 2: Effect of Testing Methods on Stress Intensification Factors,” Welding Research Bulletin 433, July 1998) showed that these measured differences should occur between cantilever and four-point tests using Markl’s method. The basic assumption in this analytical comparison is that strain-cycle correlations lead to the correct prediction of fatigue life. Using the measured alternating strain, both types of test geometries lead to the same prediction of fatigue life using these strain-cycle correlations. In this study, the same analytical assumptions used by Rodabaugh and Scavuzzo (above) are applied to a pipe where the load is varied from a four-point loading through its extremes. Loads were varied from a cantilever to an end moment by changing the dimensions of four-point loading. Also, the shape of a bilinear stress-strain curve was varied from a perfectly plastic material to various degrees of work hardening by increasing the tangent modulus in the plastic regime. The results of the study indicate that Markl’s method is conservative by predicting too short a fatigue life for four-point loading for a given stress. As indicated by this study, the differences can be very large in the low-cycle regime for a perfectly plastic material and for a four-point loading approaching an end moment. Thus, piping could be over designed with unnecessary conservatism using the current ASME Code method based on stress intensification factors.

Fatigue of glass fiber reinforced injection molded plastics. II: Tensile versus flexural loading

AbstractA previous paper reported fatigue data for several glass fiber reinforced thermoplastic materials by measuring their S‐N, stress versus number of cycles to fail, behavior. It was demonstrated that both fiber orientation and loading mode, tensile versus flexural, strongly influence the resultant S‐N data. The orientation effects are considered real, reflecting the local fiber orientation distributions in the samples, which are determined by geometry, processing and material variables. However, the higher fatigue lives observed in flexural loading are considered an artifact. Specifically for this class of materials, it is concluded that the use of linear elastic beam bending theory equations to calculate bending stresses are inappropriate. It is shown that a plasticity‐based correction factor makes the flexural S‐N data equivalent to the tensile results. In fact this correction applies equally well to monotonic loading, not just fatigue. Although it is concluded that tensile loading is preferred, some cases where flexural loading may be advantageous are indicated. Polym. Compos. 25:569–576, 2004. © 2004 Society of Plastics Engineers.

Quantum nondemolition measurement of Fock states of mesoscopic mechanical oscillators

We investigate a scheme that makes a quantum non-demolition measurement of the excitation level of a mesoscopic mechanical oscillator by utilizing the anharmonic coupling between two elastic beam bending modes. The non-linear coupling between the two modes shifts the resonant frequency of the readout oscillator proportionate to the excitation of the system oscillator. This frequency shift may be detected as a phase shift of the readout oscillation when driven on resonance. We show that in an appropriate regime this measurement approaches a quantum non-demolition measurement of the phonon number of the system oscillator. As phonon energies in micromechanical oscillators become comparable to or greater than the thermal energy, the individual phonon dynamics within each mode can be resolved. As a result it should be possible to monitor jumps between Fock states caused by the coupling of the system to the thermal reservoirs.

ANALYSIS OF CORRUGATED STEEL WEB GIRDERS BY AN EFFICIENT BEAM BENDING THEORY

An elastic beam bending theory for analysis of prestressed concrete girders with corrugated steel web is derived by the application of the variational principle. The theory is a shear deformable beam theory which is based on three displacement fields and is similar to the classical Timoshenko beam theory. A two-node linear finite element with full and reduced integration of the theory is provided. It is then used to analyze simply supported and continuous I- and box-girders. Their predicted results are found in good agreement with those by the 3D finite element analysis. A simplified theory which is similar to the proposed theory by Kato et al. (2002) is also discussed and included in appendix.

Effective stiffness modelling of composite concrete slabs in fire

In this paper a modified layered slab element is developed based on the layered procedure previously developed for the modelling of composite slabs in fire. In the development reported here, the ribs forming the lower part of any slab cast onto metal decking are included in the slab modelling. The basic idea is to use the nominal thickness of the composite slab as the thickness of the slab element, and to use an effective-stiffness factor to modify the material stiffness matrices of plain concrete in order to take into account the orthotropic properties of the slab. A maximum-strain failure criterion is used in the modelling of concrete. Three standard fire tests on metal-deck composite slabs and one full-scale natural fire test on the Cardington composite building frame are modelled for validation. The validations indicate that the model proposed is clearly capable of predicting the fire resistance of this type of composite slab in fire with reasonable accuracy. It is evident from this study that the influence of the ribs across the bottom of the slabs is significant and should be accounted for. The calculation of effective stiffness factors, which are based on the theory of elastic beam bending, is adequate and efficient, and the maximum-strain failure criterion is simple and suitable for such problems.

Effect of couple-stresses on the elastic bending of beams

The problem of the pure bending of a circular cylinder is solved within the linear couple-stress theory. The solution is obtained by correcting the classical solution with a solution in plane strain within the section. A generalized formula is thus derived for the bending inertia of a circular cross-section.

Bending of elastic beams on winkler-type viscoelastic foundations with unilateral constraints

A solution algorithm is presented for the analysis of unilateral, frictionless contact between a beam and a viscoelastic foundation. The foundation is modelled as a Winkler-type viscoelastic medium, to which the time integration method [ Mech. Teoret. i Stos. 22, 209–233 (1984)] was applied. The problem was formulated in the form of a variational inequality, from which, after space discretization by the finite element method, a linear complementarity problem was derived. The solution method covers the time-discontinuities of loads and both loading (reloading) and unloading processes. The applicability of the algorithm is illustrated by selected numerical solutions to some examples.