Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally, the finite difference approximations are derived based on fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection curve; the FOPH is made for the fourth and third derivative of the deflection curve while the SOPH is made for its second and first derivative. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, the FOPH was made for all of the derivatives of the deflection curve, and additional points were introduced at the beam’s ends and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Furthermore, tapered beams were analyzed (element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was increased through a grid refinement. Especially in the first-order analysis of uniform beams, the results were exact for uniformly distributed and concentrated loads regardless of the grid. Further research will be needed to investigate polynomial refinements (higher-order polynomials such as fifth-order, sixth-order…) of the deflection curve; the polynomial refinements aimed to increase the accuracy, whereby non-centered finite difference approximations at beam’s ends and positions of discontinuity would be used.

Design and Analysis of Cam Wave Generator Based on Free Deformation in Non-Working Area of the Flexspline

Deformation stress of a flexspline under the action of a wave generator directly affects the service life of the flexspline and meshing quality of meshing pair. This study proposed a new deformation model for a flexspline, which incorporates forced deformation in the working area and free deformation in the non-working area, keeping the deformation shape unchanged during rotating. Based on this assumption of a deformation model, the mathematical model is further established, and the design approach of a cam wave generator is developed with the deflection curve of the ring structure. Then, a sample design with a double eccentric arc cam wave generator based on this deformation model is developed and analyzed in FEM. The results show that the deformation stress of the flexspline can be improved by relaxing the forced deformation in the non-working area, and the selected deformation shape can also remain unchanged during rotating. Moreover, the stress distribution and the maximum stress value could be improved with the variation of the combination coefficient, especially for the wrap angle.

Effect of Parameters on the Design of a Suspension System with Four Oblique Springs

This paper presents the fundamental static and dynamic characteristics of a suspension system consisting of four linear springs arranged in an X-shaped configuration to achieve geometric nonlinearity. The particular interest is towards the design of a softening spring geometry realizing a quasi-zero stiffness behaviour at large deflections, and the influence of the system parameters is investigated. The static performance is studied in terms of the force-deflection curve and the dynamic performance in terms of the frequency response curve. The softening-hardening behaviour of the suspension leads to a frequency response which bends to the lower frequencies reaching a well-defined minimum. It is found that both the static and dynamic behaviours may be described in terms of a single parameter, and a simple closed-form expression is determined which links the damping in the system to the excitation amplitude to achieve the lowest possible resonance frequency.

Fatigue of structural materials in corrosive environments

The factors influencing the fatigue failure of metals and alloys are investigated. To increase the resource and reliability of products made of metals and alloys, taking into account their operation conditions and the presence of a corrosive environment, when justifying the selection of a structural material, it is proposed to take into account its deflection curve under cyclic loading. Keywords: structural material, corrosive environment, fatigue resistance, fatigue cracks, deflection curve, cyclic loading, durability. [email protected]

Numerical Simulation of the Interaction between Fibre Concrete Slab and Subsoil—The Impact of Selected Determining Factors

Shape and material optimization of building structures, including reducing the amount of concrete used, are very important aspects in sustainable construction. Numerical modelling is currently used very effectively to design optimized and sustainable structures, including their interaction with the surrounding rock environment. This paper is focused on the three selected factors of numerical modelling of fibre concrete slab and subsoil interaction: (1) the constitutive model of fibre concrete slab, (2) deformational and strength characteristics of subsoil, (3) effect of interface elements. The specialized geotechnical software Midas GTS NX, based on the finite element method, was used for the modelling of this task. Numerical results were compared with the experimental measurement of vertical displacements on the upper surface of slab. In the presented study, three constitutive models of slab recommended in MIDAS GTS NX code for modelling concrete behaviour (elastic, Mohr-Coulomb and Drucker-Prager) were applied. In addition, the sensitivity analysis with respect to the deformational and strength characteristics of subsoil was performed. The numerical study also presents the effect of the interface elements application on the slab behaviour. The numerical results of maximum vertical displacements based on the Drucker-Prager and elastic model underestimated both the experimental results and numerical results based on the Mohr-Coulomb model. From the qualitative point of view (shape of deflection curve), the numerical simulation showed the better agreement of the Mohr-Coulomb constitutive model with the experimental measurements in comparison with the other two investigated constitutive models. The performed parametric study documented that reduction of the strength and deformational characteristics of subsoil leads to the increase of maximum vertical displacements in the centre of slab, but the experimentally measured deflection curve, including uplift of slab and gapping occurrence between the slab and subsoil, was not achieved without the interface application.

Mechanical buckling of nanocomposites: Experimental and numerical investigations

The proposed research explores Multi-Walled Carbon Nanotube’s (MWCNT’s) effect on the mechanical buckling behavior of glass fiber-enhanced thermosetting composites using UTM and the load vs displacement curve is plotted. Using the inflection point method, the critical buckling load is obtained from the load vs displacement curve for beams with three different volume fractions of MWCNT. The nonlinear finite element method is used to numerically obtain the load vs deflection curve and the numerical results are compared with the experimental results, and a close match is found with the experimental results. It is observed that the nonlinearity associated with the structure can significantly reduce the critical buckling load. The critical buckling load is found to increase and reported a 27.4% increase in buckling load with 0.3 wt.% of MWCNT which could be accounted for the increase in flexural modulus of the material.

Effect of Equivalent Load Distribution on the Accuracy of Mapping the Reinforcement Load Deflection Curve in LTP

The formulations of tasks modelling embankments on soft soil, improved with columns and with reinforced load transfer platform (LTP), differ significantly. One of these differences is the distribution of equivalent load modelling part of the load carried by the LTP reinforcement and soft soil. This article analyses the influence of the nature of the load-modelling linear function, i.e., inverse triangular, uniformly distributed and triangular, as well as intermediate distributions. In total, 41 distributions of equivalent load were considered, and the results of the obtained deflection functions were compared with the measurement results of reinforcement deflection for 5 cases of experimental research available in the literature. A measure of the accuracy of mapping the reinforcement deflection curve was proposed as a relative error in relation to the deflection curve resulting from experimental measurements. Based on the analysis of the mapping error, it was determined that among the three commonly used distributions, the inverse triangular distribution shows the best fit in most of the analysed cases. However, not in every single case this is the distribution leading to a solution that best describes the behaviour of the geosynthetic reinforcement.

The Experimental Research on the Mechanical Characteristics of the Newly Designed Fuel Rod Supporting Structure of Spacer Grid

The fuel rod supporting structure of spacer grid is the key to ensure the horizontal and axial position of the fuel rod. On the purpose of improving the neutron economy and simplifying the manufacturing process of the spacer grid, the innovative curved dimple and arched spring have been designed, which could be directly punched from the strip of spacer grid. The mechanical experiments have been performed to acquire the deformation-load curves of dimples and springs on single strap and in-cell supporting structure, which could provide the load and residual deformation at 100% and 120% designed nominal deformation. It has been demonstrated that the designed in-grid cell has a relative stable load at the 100% nominal designed deformation and the test method of the single stripe is representative for the similar load-deflection curve between the spring and the in-grid cell.

Modeling and robotic handling of a snap-fitting box buckle

Modeling and robotic handling of a plastic box buckle is discussed in this paper. The closing mechanism of the box buckle is simulated to determine the characteristic of the nonlinear load-deflection curve. An intelligent end-effector was designed and manufactured to handle the assembly with a robot. The closing force is measured by a built-in load cell and its values are processed by a micro-controller. The intelligent end-effector can be used in a robotic system, which deals with different snap-fit applications.

An Enhanced Method to Evaluate Tensile Yield Stress by Small Punch Tests Using Deflection Curves

While force–displacement curves are often preferred in Small Punch (SP) tests due to the ease of the experimental set-up, they encompass significant uncertainties arising from frame compliance. In this work, a methodology is presented to predict yield stresses from the force vs. deflection curves. The present method relies on determining different force levels from the initial part of the force–deflection curve to reflect both the slope and the curvature instead of using a single force level only. The predicted yield stresses for different types of materials, that is, low- and high-strength alloys, are found to be in good agreement with the actual proof stresses with a maximum error of 16%.