The Mixed-Body Model: A Method for Predicting Large Deflections in Stepped Cantilever Beams

This paper presents a method for predicting endpoint coordinates, stress, and force to deflect stepped cantilever beams under large deflections. This method, the Mixed-Body Model or MBM, combines small deflection theory and the Pseudo-Rigid-Body Model for large deflections. To analyze the efficacy of the model, the MBM is compared to a model that assumes the first step in the beam to be rigid, to finite element analysis, and to the numerical boundary value solution over a large sample set of loading conditions, geometries, and material properties. The model was also compared to physical prototypes. In all cases, the MBM agrees well with expected values. Optimization of the MBM parameters yielded increased agreement, leading to average errors of <0.01 to 3%. The model provides a simple, quick solution with minimal error that can be particularly helpful in design.

Comparative Analysis of 3D Steel and Glulam Trusses Using ABAQUS

This study aims at developing an innovative and sustainable 3D truss system that can be applied in a variety of scenarios where intermediary propping is a hindrance. One prominent and pertinent application of such a system is in archaeological sites where long spans and no intermediary propping are desired. Previous research and studies have developed an innovative 3D steel truss for the same application. This study aims to further the innovation by providing a more sustainable alternative through Glulam and Glulam-Bamboo hybrid variations which are able to withstand similar loads as that of the steel truss but offering more sustainability and less impact on the environment as well as a light weight alternative. The results of such an alternative truss system are discussed here. One obvious problem faced with the alternative wood system was the large deflections observed and also certain regions with impermissible stresses. In addition, the proposed joint in the Glulam truss has been modelled and analyzed. It was found that the Glulam truss with lateral restraints at every quarter length of the span showed the best results in terms of deflection and stress developed. Also, the Glulam-Bamboo hybrid truss without any lateral restraints proved to be an equally effective alternative from a structural standpoint. The proposed joint system for the glulam truss also proved to be effective. The study concludes with a cost benefit analysis (CBA) between the steel, glulam and Glulam-Bamboo hybrid systems, which compares the viability of the proposed designs from an economic standpoint. The CBA shows that about 46% and 48% of costs are minimized on employing Glulam, Glulam- Bamboo respectively, instead of using Steel for the truss.

LOCALLY ORTHOTROPIC LAY-UP AS AN OPTIMAL SOLUTION FOR VAT POST-BUCKLED COMPOSITE PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE VON KARMAN LIMITS

The present paper deals with the optimization of post-buckled VAT (variable angle tow) composite plates with large deflections. The Kirchhoff assumptions are used. The plates have a symmetric lay-up. The large deflection geometrically nonlinear theory above the von Karman limits is employed. The structural potential energy is treated as a measure of structural stiffness. For the plate stiffness maximization problem, the first-order necessary conditions of the local optimality are derived. The mathematical analysis of the conditions is performed. The conditions contain two terms. One of them corresponds to the mid-plane strains; another one corresponds to the generalized plate curvatures. A locally orthotropic lay-up is identified as an optimal solution. The local ply material direction is clearly coupled with the principal directions of 2D-strains and generalized curvatures. A particular solution of the linear combination of the ply optimality conditions is indicated. For the solution two pairs of the structural tensors are co-axial: the force and the strain tensors, as well as the moment and the generalized curvature tensors.

A KINEMATIC VARIATIONAL PRINCIPLE FOR THIN METALLIC AND COMPOSITE PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE THE VON KARMAN LIMITS

Thin elastic plates (metallic or composite) experiencing large deflections are considered. The plate deflections are much larger than the plate thickness. The geometrically nonlinear elasticity theory and the Kirchhoff assumptions are employed. The elongations, the shears and the in-plane rotations are assumed to be small. A kinematic variational principle leading to a boundary value problem for the plate is derived. It is shown that the principle gives proper equilibrium equations and boundary conditions. For moderate plate deflections the principle is transformed to the case of the von Karman plate.

THIN ELASTIC PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE THE VON KARMAN LIMITS

Thin elastic plates (homogeneous or composite) experiencing large deflections are considered. The deflections are much larger than the plate thickness. The geometrically nonlinear elasticity theory and the Kirchhoff assumptions are employed. Small elongations and shears are assumed. Following Novozhilov, the strain expressions are derived. Then, under a small in-plane rotation assumption and using the virtual work principle, the equilibrium equations and the boundary conditions are obtained. The equations/conditions become the known von Karman ones for the case of moderate deflections. The solutions of the obtained equations may be used as benchmarks for the nonlinear structural analysis (e.g., FEM) software in the case of large deflections.

The Mixed-Body Model: A Method for Predicting Large Deflections in Stepped Cantilever Beams

This paper presents a method for predicting endpoint coordinates, stress, and force to deflect stepped cantilever beams under large deflections. This method, the Mixed-Body Model or MBM, combines small deflection theory and the Pseudo-Rigid-Body Model for large deflections. To analyze the efficacy of the model, the MBM is compared to a model that assumes the first step in the beam to be rigid, to finite element analysis, and to the numerical boundary value solution over a large sample set of loading conditions, geometries, and material properties. The model was also compared to physical prototypes. In all cases, the MBM agrees well with expected values. Optimization of the MBM parameters yielded increased agreement, leading to average errors of < 0.01 to 3%. The model provides a simple, quick solution with minimal error that can be particularly helpful in design.

Approximation of Physical Spline with Large Deflections

Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

Approximation of Physical Spline with Large Deflections

Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

NATURE-LIKE CURVE MODELING

A physical spline is called an elastic rod the cross- section dimensions of which are rather small as compared with the length and radius of its axis curvature. Such a rod when passing through specified points obtains in natural way a nature-like shape characterized with minimum energy of inner stresses and minimum mean curvature. A search for the equation of elastic line is a difficult mathematical problem having no elementary solution. The work purpose: the development of the experimental-rated procedure for modeling a nature-like elastic curve passing through complanar points specified in advance. The investigation methods: methods of piece-cubic interpolation based on the application of polynomial splines and compound curves specified by parametric equations. In the paper there are considered polynomial and parametric methods of the geometric modeling of the physical spline passing through the points specified in advance. The elastic line of the physical spline is obtained experimentally. The investigation results: it is shown that unlike a polynomial model a parametrized model on the basis of Fergusson curve gives high accuracy of approximation if in basic points there are specified tangents to the elastic line of the physical spline with large deflections. Novelty: there is offered a simplified method for the computation of factors of an approximating spline allowing the substitution of the 2n system of nonlinear equations (smoothness conditions) by the successive solution of n systems of two equations. Conclusions: for the modeling of nature-like curves with large deflections there is offered the application of Fergusson cubic spline passing through specified points and touching the specified straight lines in these points. The error of the modeling of the natural elastic line with free ends at n=5 does not exceed 0.4%.